\documentclass[12pt,twoside,leqno,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage{microtype}
\usepackage{amsmath,amsthm,amssymb,mathtools,mathrsfs}
\usepackage[a4paper,textwidth=13.5cm,textheight=23cm,centering]{geometry}
\usepackage{enumitem}
\usepackage{needspace}
\usepackage{xcolor}
\definecolor{paperlink}{RGB}{0,0,114}
\usepackage{hyperref}
\hypersetup{colorlinks=true,linktoc=all,linkcolor=paperlink,
  citecolor=paperlink,urlcolor=paperlink,pdfstartview=Fit}
\hypersetup{pdftitle={Relative P-Ideals of Hypergraphs and Ultrafilter Spectra},
  pdfauthor={Zhenyu Sun},
  pdfsubject={Set theory and hypergraph ideals},
  pdfkeywords={hypergraph ideal, ultrafilter, P-point, Q-point, Louveau ideal, Stone duality, reaping number}}
\pagestyle{myheadings}
\markboth{Z. Sun}{Hypergraph ideals and ultrafilter spectra}
\frenchspacing
\linespread{1.2}
\setlength{\parindent}{16pt}
\setlength{\parskip}{0pt}
\setlength{\emergencystretch}{1em}
\setcounter{tocdepth}{1}
\setlist[enumerate]{label=\textup{(\roman*)},leftmargin=2.1em,itemsep=3pt,topsep=5pt}
\allowdisplaybreaks[2]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{maintheorem}{Theorem}
\renewcommand{\themaintheorem}{\Alph{maintheorem}}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{question}[theorem]{Question}
\newtheorem{example}[theorem]{Example}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\newcommand{\om}{\omega}
\newcommand{\Fin}{\mathrm{Fin}}
\newcommand{\st}{\subseteq^*}
\newcommand{\R}{[\om]^\om}
\newcommand{\V}{\mathcal V}
\newcommand{\Lcal}{\mathcal L}
\newcommand{\Gfc}{\mathcal G_{\mathrm{fc}}}
\newcommand{\Id}{\operatorname{Id}}
\newcommand{\Star}{\operatorname{Star}}
\newcommand{\Ult}{\operatorname{Ult}}
\newcommand{\Ext}{\operatorname{Ext}}
\newcommand{\non}{\operatorname{non}}
\newcommand{\cov}{\operatorname{cov}}
\newcommand{\cof}{\operatorname{cof}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\pc}{\pi\chi}
\newcommand{\pcs}{\pi\chi_\sigma}
\newcommand{\wh}[1]{\widehat{#1}}
\newcommand{\restr}{\mathbin{\upharpoonright}}
\newcommand{\succT}{\operatorname{succ}}
\newcommand{\stem}{\operatorname{stem}}
\newcommand{\Fray}{\mathscr F_{\mathrm{fin}}}
\newcommand{\Frays}{\mathscr F_{\mathrm{fin}}^\sigma}
\newcommand{\Icap}{\mathscr I}
\newcommand{\louv}{\ell^0}
\newcommand{\set}[1]{\{#1\}}
\title{Relative $P$-Ideals of Hypergraphs and Ultrafilter Spectra}
\author{Zhenyu Sun\\[3pt]
\normalsize School of Mathematics and Statistics\\
\normalsize Northeastern University at Qinhuangdao, China\\
\normalsize\href{mailto:sunzhenyu@neuq.edu.cn}{sunzhenyu@neuq.edu.cn}}
\date{}
\begin{document}
\maketitle
\begin{abstract}
We study ideals of hypergraphs of finite uniformity under countable upper bounds
modulo finite vertex covers.  The canonical ideal generated by a free
filter and finite weak colorings has this relative $P$-property exactly
when the filter is the intersection of finitely many $P$-points.  The
classification is preserved by adjoining hypergraphs that are locally
finite after a finite vertex deletion.  The corresponding extension of a
trace ideal induces capture ideals on increasing sequences.  These are
$\sigma$-ideals exactly for $P$-filters, and their uniformities are
$\max\{\pi\chi(\mathcal F),\mathfrak d\}$.  For an ultrafilter $\mathcal U$,
the arity-two capture ideal equals the Louveau ideal exactly when
$\mathcal U$ is a $P$-point.  Successive capture ideals decrease with the
arity, and equality characterizes $Q$-points.  Every strict inclusion is
witnessed by a closed set.  Thus a non-$Q$ $P$-point yields a decreasing
sequence of $\sigma$-ideals, distinguished by closed sets, with equal
uniformities.  We also establish invariance under locally finite witness
enlargements and analyze Stone kernels and countable block products.
Consistently, nested ray-positive $P$-sets have the same singleton endpoint
spectrum and different ray transversal numbers.
\end{abstract}
\begingroup\small
\noindent\textit{2020 Mathematics Subject Classification.}
Primary 03E17; Secondary 03E05, 03E15, 05C65, 06E15, 54D80.

\smallskip
\noindent\textit{Keywords.}
Hypergraph ideal, ultrafilter, $P$-point, $Q$-point, Louveau ideal, Stone duality, reaping number.
\endgroup
\clearpage
\tableofcontents
\clearpage

\section{Introduction}

For an ideal of hypergraphs, the existence of countable upper bounds
depends on the allowed differences between a hypergraph and its bound.
We consider differences covered by finitely many vertices.  Such
differences meet any locally finite hypergraph in finitely many edges.
An increasing path avoiding a relative upper bound therefore avoids
each hypergraph it bounds.  This connects the relative $P$-property of hypergraph ideals with
countable closure of ideals on increasing sequences.

Fix an integer $k\ge2$ and write $E_k=[\om]^k$.  A hypergraph is a subset of
$E_k$.  Write $A\st B$ when $A\setminus B$ is finite.
Let $\V_k$ consist of hypergraphs with a finite vertex cover, let
$\Gfc^{(k)}$ consist of hypergraphs with a finite weak coloring, and let
$\Lcal_k$ consist of hypergraphs that become locally finite after finitely
many vertices are deleted.  A hypergraph ideal $\mathcal J$ is a
\emph{$\V_k$-$P$-ideal} if every sequence $H_n\in\mathcal J$ has an upper
bound $H\in\mathcal J$ satisfying $H_n\setminus H\in\V_k$ for all $n$.
For $R=\{r_0<r_1<\cdots\}\in\R$, the consecutive hyperedges are
\[
 P_R^{(k)}=\bigl\{\{r_n,\ldots,r_{n+k-1}\}:n<\om\bigr\}.
\]
The restriction of $\V_k$ to $P_R^{(k)}$ is the finite ideal.  This permits
upper bounds modulo $\V_k$ to control all sufficiently late consecutive
hyperedges of a witness.

For a free filter $\mathcal F$ on $\om$, put
\[
 A_X^{(k)}=E_k\setminus[X]^k,
 \qquad
 T_k(\mathcal F)=
 \{H\subseteq E_k:(\exists X\in\mathcal F)\ H\cap[X]^k\in\V_k\}.
\]
We consider the two extensions
\[
 \mathcal C_k(\mathcal F)
 =\Id\bigl(\Gfc^{(k)}\cup\{A_X^{(k)}:X\in\mathcal F\}\bigr),
 \qquad
 \mathcal J_{\mathcal F,k}=T_k(\mathcal F)+\Lcal_k.
\]
Here $\Id$ denotes the generated ideal, and $+$ denotes the ideal generated
by a pair of ideals.  The first extension incorporates finite colorings;
the second incorporates local finiteness.  Their relative $P$-properties
have different filter-theoretic descriptions.

\begin{maintheorem}\label{main:classification}
Let $\mathcal F$ be a free filter and $2\le k<\om$.
\begin{enumerate}
\item Both $\mathcal C_k(\mathcal F)$ and
$\mathcal C_k(\mathcal F)+\Lcal_k$ have the relative $P$-property precisely
when $\mathcal F$ is the intersection of a nonempty finite family of
$P$-points.
\item The ideal $\mathcal J_{\mathcal F,k}$ has the relative $P$-property
precisely when $\mathcal F$ is a $P$-filter.
\item If $\mathcal U$ is an ultrafilter, then
$\mathcal J_{\mathcal U,k}=T_k(\mathcal U)$ precisely when $\mathcal U$ is a
$Q$-point.
\end{enumerate}
\end{maintheorem}

The proofs use finite partitions with coordinatewise eventual constancy,
together with a row-tail construction for relative upper bounds.
Theorem~\ref{thm:core} proves the first assertion,
Theorem~\ref{thm:extension} the second, and
Theorem~\ref{thm:Q-absorption} the third.  The row-tail construction also
preserves the minimum size of an avoiding family when fewer than
$\mathfrak b$ locally finite hypergraphs are adjoined to a trace ideal
(Theorem~\ref{thm:small-extension}).

The hypergraphs in $\mathcal J_{\mathcal F,k}$ induce an ideal on $\R$.
Define
\begin{align*}
 \Gamma_k(H)&=\{R\in\R:|P_R^{(k)}\cap H|=\om\},\\
 \Icap_k(\mathcal F)&=
 \{\mathcal A\subseteq\R:(\exists H\in\mathcal J_{\mathcal F,k})\
       \mathcal A\subseteq\Gamma_k(H)\}.
\end{align*}
An explicit cofinal family for this ideal is given by
\begin{equation}\label{eq:intro-basis}
\begin{split}
 N_{X,f}^{(k)}={}&\{R:R\not\st X\}\\
 &{}\cup\{R\in\R:r_{n+k-1}\le f(r_n)
                         \text{ for infinitely many }n\},
\end{split}
\end{equation}
where $X\in\mathcal F$ and $f:\om\to\om$ is nondecreasing.  This description
determines both countable closure and uniformity.  Here
$\pc(\mathcal F)$ is the least size of a family of infinite sets that
refines every member of $\mathcal F$ under almost inclusion, and
$\mathfrak d$ is the dominating number.  The uniformity
$\non(\mathscr I)$ is the least size of a set outside $\mathscr I$.

\begin{maintheorem}\label{main:capture}
For every free filter $\mathcal F$ and $2\le k<\om$,
\begin{gather*}
 \Icap_k(\mathcal F)\text{ is a }\sigma\text{-ideal}
 \quad\Longleftrightarrow\quad
 \mathcal F\text{ is a }P\text{-filter},\\
 \non(\Icap_k(\mathcal F))=
       \max\{\pc(\mathcal F),\mathfrak d\}.
\end{gather*}
For a free ultrafilter $\mathcal U$, the ideals satisfy
\[
 \Icap_{k+1}(\mathcal U)\subseteq\Icap_k(\mathcal U),
 \qquad
 \Icap_{k+1}(\mathcal U)=\Icap_k(\mathcal U)
 \quad\Longleftrightarrow\quad
 \mathcal U\text{ is a }Q\text{-point}.
\]
If $\mathcal U$ is not a $Q$-point, each strict inclusion is witnessed
by a closed subset of $\R$ belonging to $\Icap_k(\mathcal U)$ and not
to $\Icap_{k+1}(\mathcal U)$.
Moreover,
\[
 \Icap_2(\mathcal U)=\louv_{\mathcal U}
 \quad\Longleftrightarrow\quad
 \mathcal U\text{ is a }P\text{-point},
\]
where $\louv_{\mathcal U}$ is the Louveau ideal.  For every $k\ge3$,
$\Icap_k(\mathcal U)=\louv_{\mathcal U}$ holds precisely when $\mathcal U$
is both a $P$-point and a $Q$-point.
\end{maintheorem}

Consequently, a non-$Q$ $P$-point yields
\[
 \louv_{\mathcal U}=\Icap_2(\mathcal U)
 \supsetneq\Icap_3(\mathcal U)
 \supsetneq\Icap_4(\mathcal U)\supsetneq\cdots,
\]
a strictly decreasing sequence of $\sigma$-ideals with the same uniformity.
The separating closed sets are defined by finite blocks from a partition
witnessing failure of the $Q$-point property.
At a fixed arity, the ideal is unchanged if $P_R^{(k)}$ is enlarged to any
locally finite subhypergraph of $[R]^k$.  Thus the arity and the choice of
locally finite witness enlargement have distinct effects.

Theorem~\ref{thm:basis} proves the basis description and countable-closure
criterion, and Theorem~\ref{thm:cardinals} computes the uniformity.
The Louveau identification and the arity classification are proved in
Theorems~\ref{thm:louveau-identification} and~\ref{thm:arity}.

The Stone space $X_k=\Ult(\mathcal P(E_k)/\V_k)$ gives a topological
interpretation.  Its points are ultrafilters on $E_k$ disjoint from
$\V_k$.  The $k$ coordinate maps of an increasing hyperedge induce
continuous maps $\pi_j:X_k\to\om^*$, where $\om^*$ is the space of
free ultrafilters on $\om$.  Let $Z_k$ be their equalizer and write
$\pi$ for the common map on $Z_k$.  For a proper ideal
$\mathcal J\supseteq\Gfc^{(k)}$, its Stone kernel
$L_{\mathcal J}=\{p\in X_k:p\cap\mathcal J=\varnothing\}$ lies in
$Z_k$, and its endpoint spectrum is $\pi[L_{\mathcal J}]$.

A closed set is a $P$-set if every countable intersection of its
neighborhoods is again a neighborhood.  We call a nonempty closed
$L\subseteq X_k$ ray-positive if every clopen neighborhood of $L$
contains $\wh{P_R^{(k)}}$ for some $R\in\R$; here $\wh H$ is the
clopen set of ultrafilters containing $H$.  Its ray transversal number
$\nu_k(L)$ is the least size of a family of rays that supplies such a
witness in every clopen neighborhood.  A ray $R$ avoids $H$ if
$|P_R^{(k)}\cap H|<\om$, and an ideal is ray-proper if each of its members
is avoided by some ray.  We write $\beta\om$ for the space of all
ultrafilters on $\om$, including the principal ones; a function
$e:\om\to\om$ induces the map
$\beta e(\mathcal U)=\{A\subseteq\om:e^{-1}[A]\in\mathcal U\}$.

\begin{maintheorem}\label{main:stone}
\begin{enumerate}
\item For a proper ideal $\mathcal J\supseteq\Gfc^{(k)}$,
$L_{\mathcal J}$ is a $P$-set in $X_k$ if and only if $\mathcal J$ is a
$\V_k$-$P$-ideal.  If a nonempty $C\subseteq Z_k$ is a $P$-set in $X_k$
and $\pi[C]=\{\mathcal U\}$, then $\mathcal U$ is a $P$-point.
\item Let $\om=\dot\bigcup_n B_n$ be a partition into infinite sets,
and on each $[B_n]^k$ let $\mathcal J_n$ be a proper ray-proper
relative $P$-ideal containing the finite-chromatic ideal.  Then
\begin{equation*}
 \mathcal J=\{H\subseteq E_k:H\cap[B_n]^k\in\mathcal J_n
                                      \text{ for every }n\}
\end{equation*}
has the same properties.  On its kernel $L_{\mathcal J}$, the
block-index function $e(i)=n$ for $i\in B_n$ induces a continuous
surjection $\rho=\beta e\circ\pi:L_{\mathcal J}\to\beta\om$.
Every fiber is ray-positive.  Every endpoint of a point in a
nonprincipal fiber fails to be a $P$-point.  Every nonempty $P$-set
in the relative topology of $L_{\mathcal J}$ meets a principal fiber.
\item If $\mathcal U$ is a $P$-point with
$\pc(\mathcal U)<\mathfrak d$, the kernels
$L_{\mathcal J_{\mathcal U,k}}$ and $L_{T_k(\mathcal U)}$ are
ray-positive $P$-sets in $X_k$ with
endpoint spectrum $\{\mathcal U\}$ and respective ray transversal
numbers $\mathfrak d$ and $\pc(\mathcal U)$.  They satisfy
\begin{equation*}
 L_{\mathcal J_{\mathcal U,k}}\subsetneq L_{T_k(\mathcal U)}.
\end{equation*}
\end{enumerate}
\end{maintheorem}

The first and third assertions follow from the Stone correspondence and
the filter classifications in Section~\ref{sec:stone}.  The product and
fiber assertions are proved in Section~\ref{sec:blocks}.

The relevant antecedents are the finite-chromatic ideal and pair-reaping
studied by Hru\v{s}\'ak, Meza-Alc\'antara, and Minami \cite{HMM2010}, and the
ultrafilter ideals studied by Brendle and Shelah \cite{BS1999}.  The former
work treats unbounded families of finite tuples, including higher finite
arities, and the associated splitting invariants.  The present paper uses
overlapping consecutive hyperedges, finite vertex incidence, and upper
bounds modulo finite vertex covers.  Brendle and Shelah proved
\[
 \non(\louv_{\mathcal U})=
       \max\{\pcs(\mathcal U),\mathfrak d\}.
\]
For $P$-points, the uniformity formula in Theorem~\ref{main:capture}
recovers this value through an explicit hypergraph representation.  We
identify when that representation coincides with the Louveau ideal and
classify the higher-arity capture ideals by strict inclusion.  The latter
inclusions are already separated by closed sets.  The Louveau ideal itself originates in
\cite{Louveau1976}; its successor-set description is recalled and proved
in Section~\ref{sec:louveau}.

Countable decision on $\om$ also connects these ideals to reaping.
The dual of the countably complete decision filter is an instance of
Schilhan's ideal associated with the refining relation
\cite[Definitions~2.9 and~7.6]{Schilhan2026}.
Theorem~\ref{thm:absolute} characterizes finite-chromatic hypergraphs
by the existence of a countable family of subsets of $\om$ whose
simultaneous decision ensures avoidance.  Section~\ref{sec:reaping}
uses this characterization to represent $\mathfrak r$ and
$\mathfrak r_\sigma$ by rays and to formulate ideal-extension questions.

\section{Hypergraphs and increasing sequences}\label{sec:prelim}

Finite vertex covers give an eventual inclusion order on the rows of a
hypergraph.  We establish this description, relate finite colorings to
decisions of subsets of $\om$, and construct countable upper bounds by
truncating rows.

\subsection{Filters and cardinal characteristics}

We work in ZFC.  The set $\om$ consists of the nonnegative integers.
For bounds on finite subsets of $\om$, we use $\max\varnothing=-1$.  For a
set $X$, write $[X]^k$, $[X]^{<\om}$, and $[X]^\om$ for its $k$-element,
finite, and countably infinite subsets.  We write $A\st B$ when
$A\setminus B$ is finite.  A filter is proper, closed under finite
intersections and supersets.  Unless stated otherwise, filters on $\om$ are
free, so they contain all cofinite sets and have no finite member.  An ultrafilter
is a maximal proper filter; equivalently, it contains exactly one of $X$
and $\om\setminus X$ for every $X\subseteq\om$.  Every proper filter extends
to an ultrafilter by Zorn's lemma.

A \emph{pseudointersection} of a family $\mathcal A\subseteq\R$ is an
infinite set $B$ such that $B\st A$ for every $A\in\mathcal A$.  A free
filter $\mathcal F$ is a \emph{$P$-filter} if each countable subfamily has a
pseudointersection belonging to $\mathcal F$.  A $P$-point is a free
ultrafilter that is a $P$-filter.  A free ultrafilter $\mathcal U$ is a
\emph{$Q$-point} if, for every partition $\om=\dot\bigcup_n C_n$ into finite
sets, some $X\in\mathcal U$ satisfies $|X\cap C_n|\le1$ for all $n$.

A family $\mathcal B\subseteq\R$ is a \emph{$\pi$-base} of $\mathcal F$ if
for every $X\in\mathcal F$ there is $B\in\mathcal B$ with $B\st X$.  It is
a \emph{$\pi_\sigma$-base} if, for every sequence $X_n\in\mathcal F$, one
$B\in\mathcal B$ satisfies $B\st X_n$ for every $n$.  The least respective
sizes are $\pc(\mathcal F)$ and $\pcs(\mathcal F)$.  The members of these
bases are arbitrary infinite subsets of $\om$.  A filter base is a family
$\mathcal B\subseteq\mathcal F$ such that every $X\in\mathcal F$ almost
contains some $B\in\mathcal B$.  Its least size is the
\emph{character} $\chi(\mathcal F)$.  Closing a base under finite deletions
turns it into a base under actual inclusion without changing an infinite
cardinality.  These conventions agree with \cite{BS1999}.

Every countable subfamily of a free filter has an infinite
pseudointersection.  Choose its $n$th point above the preceding point in the
intersection of its first $n+1$ members.  Thus $\pcs(\mathcal F)$ is always
defined.  If $\mathcal F$ is a $P$-filter, a $\pi$-base refines the
pseudointersection in the filter, so
\begin{equation}\label{eq:pi-P}
 \pcs(\mathcal F)=\pc(\mathcal F).
\end{equation}
A free ultrafilter has no countable $\pi$-base.  Indeed, a countable family of
infinite sets can be split by one set, by successively choosing fresh points
on both sides from each member.  Whichever side belongs to the ultrafilter
then contains no member of the alleged $\pi$-base modulo finite.

For functions $f,g\in\om^\om$, write $f\le^*g$ if $f(n)\le g(n)$ for all
but finitely many $n$.  The bounding number $\mathfrak b$ is the least size
of a $\le^*$-unbounded family, and the dominating number $\mathfrak d$ is the
least size of a family cofinal under $\le^*$.  We use nondecreasing majorants
freely.  A dominating family may be chosen to consist of functions $h$ with
$h(n)>n$.  For background on these cardinals, see
\cite{Blass2010}.

For a proper ideal $\mathscr I$ on a set $S$, its
\emph{uniformity} is
\[
 \non(\mathscr I)=\min\{|A|:A\subseteq S,\ A\notin\mathscr I\}.
\]
For a proper filter $\mathscr F$ on $S$, its transversal number is
\[
 \tr(\mathscr F)=\min\{|A|:A\subseteq S,\ A\cap F\ne\varnothing
                                      \text{ for all }F\in\mathscr F\}.
\]
The complement-dual ideal $\{A\subseteq S:S\setminus A\in\mathscr F\}$ has
uniformity $\tr(\mathscr F)$, since $A$ fails to meet a filter member exactly
when its complement belongs to the filter.

\subsection{Finite vertex covers and local finiteness}

Fix $2\le k<\om$.  A \emph{$k$-uniform hypergraph} on $\om$ is a subset
$H\subseteq E_k=[\om]^k$.  Its elements are its hyperedges.  A set $B$ is
\emph{independent} if $H\cap[B]^k=\varnothing$.  A weak coloring is a map
$c:\om\to q$ such that no hyperedge is monochromatic.  A finite weak
coloring partitions the vertex set into finitely many independent sets.
We write $\chi(H)$ for the least number of colors and put
\[
 \Gfc^{(k)}=\{H\subseteq E_k:\chi(H)<\om\}.
\]
This is an ideal.  For a union of two finitely colorable hypergraphs, the
ordered pair of their colorings is a finite weak coloring.  Throughout,
hypergraph coloring has this weak meaning.

For $F\subseteq\om$, put
\[
 \Star_k(F)=\{e\in E_k:e\cap F\ne\varnothing\}.
\]
The finite-vertex-cover ideal is
\[
 \V_k=\{H\subseteq E_k:(\exists F\in[\om]^{<\om})\ H\subseteq\Star_k(F)\}.
\]
It is contained in $\Gfc^{(k)}$.  Assign distinct colors to the finitely many
covering vertices and one further color to all remaining vertices.
For $i\in\om$, define the $i$th row by $H(i)=\{e\in H:\min e=i\}$.

\begin{lemma}[Row description]\label{lem:rows}
For $H,K\subseteq E_k$, the following are equivalent:
\begin{enumerate}
\item $H\setminus K\in\V_k$;
\item $H(i)\subseteq K(i)$ for all but finitely many $i$;
\item for some $N$, every $e\in H$ with $\min e\ge N$ belongs to $K$.
\end{enumerate}
\end{lemma}
\begin{proof}
If a finite set $F$ covers $H\setminus K$, no error hyperedge can have
minimum larger than every member of $F$.  This proves (i)$\Rightarrow$(iii).
The equivalence of (ii) and (iii) is immediate.  Under (iii), every error
hyperedge meets $[0,N)$, proving (i).
\end{proof}

A hypergraph is \emph{locally finite} if every vertex belongs to finitely many
hyperedges.  Let $\Lcal_k$ consist of hypergraphs $H$ for which
$H\cap[\om\setminus F]^k$ is locally finite for some finite $F$.

\begin{lemma}\label{lem:LF-rows}
A hypergraph belongs to $\Lcal_k$ exactly when all but finitely many of its
rows are finite.  Consequently, $\Lcal_k$ is an ideal containing $\V_k$.
Every infinite vertex set contains an infinite independent subset for any
member of $\Lcal_k$.
\end{lemma}
\begin{proof}
If deleting $F$ gives local finiteness, every row with minimum above $\max F$
is finite.  Conversely, suppose all rows with minimum at least $N$ are
finite.  In the induced hypergraph on $[N,\infty)$, the hyperedges incident
with a fixed vertex $v$ lie in the finite union of rows indexed by
$N,\ldots,v$, so there are finitely many of them.  The ideal assertions follow
from this row description.

For the final assertion, first delete the finite exceptional vertices.  Given
finitely many already selected vertices, their incident hyperedges involve
only finitely many vertices.  Choose the next vertex in the prescribed
infinite set outside that finite union and above the previous choice.  No
hyperedge can be completed by a later choice.  The resulting infinite set
is independent.
\end{proof}

After identifying each row with $\om$, the ideal $\Lcal_k$ becomes
$\Fin\otimes\Fin=\{A\subseteq\om\times\om:
\{n:A_n\text{ is infinite}\}\text{ is finite}\}$.
Our upper bounds use the error ideal $\V_k$, whose members have finitely
many nonempty rows.

Write $\mathcal M_k$ for the nondecreasing functions $f\in\om^\om$ with
$f(i)\ge i+k$, and set
\begin{equation}\label{eq:bounded-hypergraph}
 B_f^{(k)}=\{e\in E_k:\max e\le f(\min e)\}.
\end{equation}

\begin{lemma}[Bounded-row hypergraphs]\label{lem:Bf}
For every $H\in\Lcal_k$ there is $f\in\mathcal M_k$ such that
$H\setminus B_f^{(k)}\in\V_k$.  Moreover,
\[
 B_f^{(k)}\setminus B_g^{(k)}\in\V_k
 \quad\Longleftrightarrow\quad f\le^*g
 \qquad(f,g\in\mathcal M_k).
\]
\end{lemma}
\begin{proof}
Above the finitely many exceptional rows, take the maximum of all vertices
in each finite row and then a nondecreasing majorant.  This gives the first
assertion by Lemma~\ref{lem:rows}.
If $f\le^*g$, row inclusion holds eventually.  If $f(i)>g(i)$, the set
\[
 \{i,i+1,\ldots,i+k-2,g(i)+1\}
\]
is a $k$-element hyperedge in the $i$th row of the difference.  Infinitely
many such indices preclude a finite vertex cover, again by
Lemma~\ref{lem:rows}.
\end{proof}

\subsection{Witnesses and absolute avoidance}

We regard $\R$ as the space of its increasing enumerations, with the
subspace topology inherited from $\om^\om$.  The first finitely many entries
specify a clopen cylinder.  For each $R=\{r_0<r_1<\cdots\}$, let
\[
 P_R^{(k)}=\{\{r_n,\ldots,r_{n+k-1}\}:n<\om\}.
\]
A witness assignment $W$ used below satisfies
\begin{equation}\label{eq:W}
 P_R^{(k)}\subseteq W(R)\subseteq[R]^k,
 \qquad W(R)\text{ is locally finite}
 \quad(R\in\R).
\end{equation}
Set
\[
 D_H^W=\{R:|W(R)\cap H|<\om\},\qquad
 \Gamma_W(H)=\R\setminus D_H^W.
\]
When $W(R)=P_R^{(k)}$, we write $D_H^{(k)}$ and $\Gamma_k(H)$.
We abbreviate $D_H^{(k)}$ to $D_H$ when the arity is fixed.
A hypergraph ideal is \emph{$W$-ray-proper} if $D_H^W\ne\varnothing$ for
every member $H$.  The term \emph{ray-proper} refers to consecutive
hyperedges.
A family $\mathcal A\subseteq\R$ \emph{avoids} an ideal $\mathcal J$ if
for every $H\in\mathcal J$ some $R\in\mathcal A$ belongs to $D_H^W$.
Thus the avoiding ray may depend on $H$.

\begin{lemma}[The witness null ideal]\label{lem:null-ideal}
For every assignment satisfying \eqref{eq:W},
\[
 \Gamma_W(H)=\varnothing\quad\Longleftrightarrow\quad H\in\V_k.
\]
Furthermore,
\[
 \Gamma_W(H\cup K)=\Gamma_W(H)\cup\Gamma_W(K),
 \qquad D_{H\cup K}^W=D_H^W\cap D_K^W,
\]
and $H\triangle K\in\V_k$ implies $\Gamma_W(H)=\Gamma_W(K)$.
\end{lemma}
\begin{proof}
A finite vertex cover meets finitely many hyperedges of the locally finite
witness.  Thus every member of $\V_k$ has empty capture set.  If
$H\notin\V_k$, recursively choose $e_n\in H$ with
$\max e_n<\min e_{n+1}$.  A new choice is possible because an initial finite
interval does not cover $H$.  In the increasing enumeration of
$R=\bigcup_ne_n$, each $e_n$ is a consecutive block of length $k$.
Hence $W(R)\cap H$ is infinite.  The remaining identities follow from finite
union and from the first assertion.
\end{proof}

\begin{example}\label{ex:witnesses}
For any fixed $d\ge k-1$, taking all $k$-subsets of
$\{r_n,\ldots,r_{n+d}\}$, for every $n$, gives an assignment satisfying
\eqref{eq:W}.  For $k=2$, the assignment
\[
 W(R)=P_R^{(2)}\cup
 \{\{r_n,r_{2n+1}\},\{r_n,r_{2n+2}\}:n<\om\}
\]
also satisfies \eqref{eq:W}.  Every index has at most three incidences in
the added binary-tree graph and at most two in the consecutive path.
The complete hypergraph $[R]^k$ has infinite vertex degrees and lies outside
\eqref{eq:W}.
\end{example}

For $X\subseteq\om$, put
\[
 \Delta_kX=\{e\in E_k:0<|e\cap X|<k\}.
\]
This hypergraph has a two-color weak coloring.
A set $R\in\R$ \emph{decides} $X$ if $R\st X$ or $R\st X^c$, where
$X^c=\om\setminus X$.

\begin{lemma}[Cuts and decisions]\label{lem:cuts}
For every $R\in\R$ and every witness assignment satisfying \eqref{eq:W},
\[
 R\in D_{\Delta_kX}^W
 \quad\Longleftrightarrow\quad R\st X\text{ or }R\st X^c.
\]
Also, $D_{A_X^{(k)}}^W=\{R:R\st X\}$.  Every member of
$\Gfc^{(k)}$ is contained in a finite union of hypergraphs $\Delta_kX$.
\end{lemma}
\begin{proof}
If all sufficiently late consecutive $k$-blocks are monochromatic for
$\mathbf1_X$, their overlaps force all sufficiently late entries to have one
common value.  Conversely, an eventually monochromatic sequence has mixed
witness hyperedges only through its finitely many exceptional vertices,
which have finite incidence.  If $R\setminus X$ is infinite, infinitely many
consecutive blocks meet $R\setminus X$; this proves the assertion for
$A_X^{(k)}$.

For the final assertion, encode the finitely many colors of a weak coloring
by distinct binary strings.  Each hyperedge has two different colors, and
some binary coordinate separates them.  The cuts associated with those
coordinates cover the hypergraph.
\end{proof}

\begin{theorem}[Absolute avoidance and finite weak colorings]
\label{thm:absolute}
For $H\subseteq E_k$ and $W$ satisfying \eqref{eq:W}, the following are
equivalent:
\begin{enumerate}
\item $H\in\Gfc^{(k)}$;
\item there are sets $X_n\subseteq\om$ such that every $R$ deciding all
$X_n$ belongs to $D_H^W$.
\end{enumerate}
\end{theorem}
\begin{proof}
If $H$ has a finite weak coloring, a set deciding every color class is almost
contained in one class.  All hyperedges of $W(R)\cap H$ then meet finitely
many exceptional vertices, so there are only finitely many.

Conversely, suppose $\chi(H)=\om$ and fix $X_n\subseteq\om$.  Construct
$C_0\supseteq C_1\supseteq\cdots$ with $C_0=\om$,
$\chi(H\cap[C_n]^k)=\om$, and $C_{n+1}$ contained in one side of $X_n$.
If both sides had finite weak colorings, assigning disjoint finite palettes
to them would give a finite weak coloring of $H\cap[C_n]^k$.
Deleting finitely many vertices also preserves infinite weak chromatic
number, since deleted vertices could otherwise receive separate colors.
We can therefore choose $e_n\in H\cap[C_n]^k$ with
$\max e_n<\min e_{n+1}$.  The set $R=\bigcup_ne_n$ decides every $X_n$,
because its tail is contained in $C_{n+1}$.  Each $e_n$ is a consecutive
$k$-block of $R$, so $W(R)\cap H$ is infinite.  This contradicts (ii).
\end{proof}

For the consecutive assignment, $\Gamma_k(H)$ is a $G_\delta$ subset of
$\R$.  It is the intersection over $m$ of the union over $n\ge m$ of the
clopen conditions $\{r_n,\ldots,r_{n+k-1}\}\in H$.


\subsection{Row-tail bounds and local finiteness}\label{subsec:row-bounds}

For hypergraphs $G,H\subseteq E_k$, call $H$ a relative upper bound of $G$
when $G\setminus H\in\V_k$.  An ideal has the relative $P$-property if every
countable subfamily has such an upper bound in the ideal.  The next lemma
reduces arbitrary relative upper bounds to explicit truncations of rows.

Let $G_0\subseteq G_1\subseteq\cdots$ be hypergraphs on $E_k$.
An arbitrary sequence in an ideal is reduced to this form by taking its
successive finite unions.  For $R\in\bigcap_nD_{G_n}$ define
\[
 \lambda_R(n)=
 \begin{cases}
 0,&P_R^{(k)}\cap G_n=\varnothing,\\
 1+\max\{\min e:e\in P_R^{(k)}\cap G_n\},&
       P_R^{(k)}\cap G_n\neq\varnothing.
 \end{cases}
\]
For a strictly increasing $h:\om\to\om$, set
\begin{equation}\label{eq:general-row-tail}
 H_h=\bigcup_n\{e\in G_n:\min e\geq h(n)\}.
\end{equation}

\begin{lemma}\label{lem:row-normal-form}
Every $H_h$ in \eqref{eq:general-row-tail} is a common
$\V_k$-upper bound of the $G_n$, and
\begin{equation}\label{eq:row-avoidance}
 D_{H_h}=
 \{R\in\bigcap_nD_{G_n}:\lambda_R\leq^*h\}.
\end{equation}
Every common $\V_k$-upper bound $H$ contains some $H_h$.
\end{lemma}
\begin{proof}
The difference $G_n\setminus H_h$ is covered by $[0,h(n))$.
Thus any ray avoiding $H_h$ also avoids every $G_n$.
If $\lambda_R\leq^*h$, then for all but finitely many $n$ no ray edge in
$G_n$ has minimum at least $h(n)$.  The remaining finitely many $G_n$
each meet the ray finitely, so $R$ avoids $H_h$.
If $\lambda_R(n)>h(n)$ for infinitely many $n$, choose corresponding ray
edges in $G_n$ with minimum at least $h(n)$.  Since $h(n)\to\infty$,
infinitely many distinct edges are obtained, all belonging to $H_h$.
This proves \eqref{eq:row-avoidance}.
Finally, if $G_n\setminus H\in\V_k$, Lemma~\ref{lem:rows} gives a
threshold $a_n$ above which every row of $G_n$ is contained in $H$.
Choose a strictly increasing $h$ with $h(n)\geq a_n$.  Then $H_h\subseteq H$.
\end{proof}


\begin{lemma}[Locally finite pseudounions]\label{lem:LF-P}
The ideal $\Lcal_k$ is a $\V_k$-$P$-ideal.  More precisely, for any sequence
$H_n\in\Lcal_k$ there is a locally finite $H$ with
$H_n\setminus H\in\V_k$ for every $n$.
\end{lemma}
\begin{proof}
Choose $a_n$ such that the $i$th row of $H_n$ is finite whenever $i\ge a_n$.
Take a strictly increasing function $h$ with $h(n)\ge a_n$, and define
\begin{equation}\label{eq:row-tail-LF}
 H=\bigcup_{n<\om}\{e\in H_n:\min e\ge h(n)\}.
\end{equation}
For fixed $i$, only finitely many $n$ satisfy $h(n)\le i$.  For each of
these $n$, the row $H_n(i)$ is finite.  Hence every row of $H$ is finite,
which implies local finiteness by the proof of Lemma~\ref{lem:LF-rows}.
Moreover, $H_n\setminus H$ is covered by $[0,h(n))$.
\end{proof}

The ideal $\Lcal_k$ fails the ordinary $P$-ideal property modulo finite
sets of hyperedges.  Each full row belongs to $\Lcal_k$, whereas an upper
bound modulo finite hyperedges of all full rows would have infinitely
many infinite rows.

\section{Canonical ideals generated by filters}\label{sec:canonical}

The trace ideal records the sets in a free filter as independent vertex
sets.  Adjoining the finite-chromatic ideal imposes an additional
requirement on countable upper bounds.  We characterize this requirement
by finite partitions and then by the ultrafilters extending the filter.

\subsection{Trace ideals and canonical filters}

An ideal on $E_k$ is downward closed and closed under finite unions.  All
hypergraph ideals considered below contain $\V_k$.  Properness means
$E_k\notin\mathcal J$.  We write
\[
 \mathcal I+\mathcal J
 =\{H\subseteq E_k:(\exists I\in\mathcal I)(\exists J\in\mathcal J)
                                      \ H\subseteq I\cup J\}.
\]
The sum is the ideal generated by $\mathcal I\cup\mathcal J$.

\begin{definition}
For a free filter $\mathcal F$, its trace ideal is
\[
 T_k(\mathcal F)=
 \{H\subseteq E_k:(\exists X\in\mathcal F)\ H\cap[X]^k\in\V_k\}.
\]
For a proper ideal $\mathcal J$ containing $\V_k$, its canonical filter is
\[
 \mathcal F_{\mathcal J}
   =\{X\subseteq\om:A_X^{(k)}\in\mathcal J\},
 \qquad A_X^{(k)}=E_k\setminus[X]^k.
\]
\end{definition}

\begin{proposition}[Trace ideals]\label{prop:trace}
Let $\mathcal F$ be a free filter.  Then $T_k(\mathcal F)$ is a proper
$W$-ray-proper ideal for every $W$ satisfying \eqref{eq:W}, and
\[
 T_k(\mathcal F)=\{H:(\exists X\in\mathcal F)\ H\subseteq A_X^{(k)}\},
 \qquad
 \mathcal F_{T_k(\mathcal F)}=\mathcal F.
\]
Furthermore,
\begin{enumerate}
\item $T_k(\mathcal F)$ is a $\V_k$-$P$-ideal if and only if $\mathcal F$
is a $P$-filter;
\item $\Gfc^{(k)}\subseteq T_k(\mathcal F)$ if and only if $\mathcal F$ is
an ultrafilter.
\end{enumerate}
\end{proposition}
\begin{proof}
Suppose $H\cap[X]^k$ is covered by a finite set $F$.  Then
$Y=X\setminus F\in\mathcal F$ and $H\cap[Y]^k=\varnothing$, so
$H\subseteq A_Y^{(k)}$.  The reverse inclusion is immediate.  The identity
\begin{equation}\label{eq:AX-intersection}
 A_{X\cap Y}^{(k)}=A_X^{(k)}\cup A_Y^{(k)}
\end{equation}
shows that $T_k(\mathcal F)$ is an ideal.  An infinite subset of $Y$ is
independent for $H$, proving properness and $W$-ray-properness.
If $A_Z^{(k)}\subseteq A_Y^{(k)}$ for some infinite $Y\in\mathcal F$, then
$Y\subseteq Z$.  A point of $Y\setminus Z$, together with $k-1$ other points
of $Y$, would contradict the inclusion.  Thus the canonical filter is
exactly $\mathcal F$.

Assume $\mathcal F$ is a $P$-filter and $H_n\subseteq A_{X_n}^{(k)}$ with
$X_n\in\mathcal F$.  Choose $X\in\mathcal F$ with $X\st X_n$ for all $n$.
Then
\[
 H_n\setminus A_X^{(k)}
 \subseteq A_{X_n}^{(k)}\cap[X]^k
 \subseteq\Star_k(X\setminus X_n)\in\V_k.
\]
Conversely, let $X_n\in\mathcal F$ and take a common relative upper bound
$H\in T_k(\mathcal F)$ for the $A_{X_n}^{(k)}$.  Choose $Y\in\mathcal F$
with $H\cap[Y]^k=\varnothing$.  For a finite cover $F_n$ of
$A_{X_n}^{(k)}\setminus H$, an infinite $Y\setminus X_n$ would contain
$k$ points outside $F_n$.  Their hyperedge belongs to the difference and
misses its cover.  Hence $Y\st X_n$ for all $n$.

If $\mathcal F$ is an ultrafilter, it contains one cell of every finite
coloring, so it contains an independent set for every member of
$\Gfc^{(k)}$.  Conversely, if every $\Delta_kX$ belongs to the trace ideal,
there is $Y\in\mathcal F$ with $\Delta_kX\cap[Y]^k=\varnothing$.
Since $Y$ is infinite, it is contained in $X$ or in $X^c$.  Thus
$\mathcal F$ decides every subset of $\om$ and is an ultrafilter.
\end{proof}

\begin{lemma}[Canonical filter]\label{lem:canonical-filter}
If $\mathcal J$ is a proper ideal containing $\V_k$, then
$\mathcal F_{\mathcal J}$ is a free filter.  If in addition
$\Gfc^{(k)}\subseteq\mathcal J$ and $\mathcal J$ is a $\V_k$-$P$-ideal,
then $\mathcal F_{\mathcal J}$ is a $P$-filter.
\end{lemma}
\begin{proof}
We have $A_\om^{(k)}=\varnothing$.  If a finite $X$ belonged to the
canonical filter, both $A_X^{(k)}$ and $[X]^k$ would belong to $\mathcal J$,
contradicting properness.  Equation~\eqref{eq:AX-intersection} gives finite
intersection closure.  If $X\st Y$, then
\[
 A_Y^{(k)}\setminus A_X^{(k)}\subseteq\Star_k(X\setminus Y),
\]
which proves upward closure modulo finite and includes all cofinite sets.

For the $P$-property, replace a given sequence by its successive finite
intersections and write it as $X_0\supseteq X_1\supseteq\cdots$.
Choose $H\in\mathcal J$ and finite $F_n$ such that
\[
 A_{X_n}^{(k)}\setminus H\subseteq\Star_k(F_n).
\]
For a vertex $v$, define its exit level
\[
 a(v)=\min\{n:v\notin X_n\}\in\om\cup\{\infty\},
\]
where the minimum of the empty set is $\infty$, and put
\begin{equation}\label{eq:exit-set}
 Y=\{v:a(v)=\infty\}
   \cup\bigcup_{n<\om}\{v\in F_n:a(v)\ge n\}.
\end{equation}
If $v\in Y\setminus X_m$, its exit level is at most $m$, and therefore it
belongs to some $F_n$ with $n\le m$.  Hence
\[
 Y\setminus X_m\subseteq\bigcup_{n\le m}F_n,
 \qquad Y\st X_m.
\]
We claim $[Y^c]^k\subseteq H$.  Otherwise take $e\in[Y^c]^k\setminus H$ and
let $n=\min\{a(v):v\in e\}$.  All exit levels are finite, and
$e\in A_{X_n}^{(k)}\setminus H$.  Some $w\in e\cap F_n$ then satisfies
$a(w)\ge n$, so \eqref{eq:exit-set} puts $w$ in $Y$, a contradiction.
Finally,
\[
 A_Y^{(k)}=[Y^c]^k\cup\Delta_kY\in\mathcal J.
\]
Thus $Y\in\mathcal F_{\mathcal J}$, and properness ensures that $Y$ is
infinite.
\end{proof}

\begin{proposition}[Infinite complete subhypergraphs]\label{prop:tree-clique}
Suppose $\mathcal J\supseteq\Gfc^{(k)}$ is a $\V_k$-$P$-ideal.  Every
infinite $X\subseteq\om$ contains an infinite $B$ with $[B]^k\in\mathcal J$.
In particular, $\Gfc^{(k)}$ is not a $\V_k$-$P$-ideal.
\end{proposition}
\begin{proof}
Let $T=\{s\in\om^{<\om}:|s|\ge1\}$ and choose distinct labels $x_s\in X$.
For $n<\om$, let $G_n$ consist of the hyperedges
\[
 \{x_{s_0},\ldots,x_{s_{k-1}}\},
 \qquad
 s_0\subsetneq\cdots\subsetneq s_{k-1},\quad |s_0|=n+1.
\]
Color the labels at level $n+1$ with one color and all other vertices with
another.  Each edge of $G_n$ has exactly one vertex of the first color, so
$G_n\in\Gfc^{(k)}$.
Choose $H\in\mathcal J$ and finite $F_n$ covering $G_n\setminus H$.
Select a branch $s_0\subsetneq s_1\subsetneq\cdots$, with $|s_n|=n+1$, so
that $x_{s_n}\notin\bigcup_{j\le n}F_j$.  At the first step there are
infinitely many roots; at each later step there are infinitely many
immediate successors.  Thus every choice avoids the finite forbidden set.
If $n_0<\cdots<n_{k-1}$, the corresponding edge lies in $G_{n_0}$ and misses
$F_{n_0}$, so it lies in $H$.  Therefore
$B=\{x_{s_n}:n<\om\}$ satisfies $[B]^k\subseteq H$.  An infinite complete
$k$-uniform hypergraph has no finite weak coloring, proving the last
assertion.
\end{proof}

\subsection{Finite ultrafilter spectra}

Compactness turns the condition that every extending ultrafilter is a
$P$-point into finiteness of the family of extensions.
Identify $\beta\om$ with the ultrafilters on $\om$, with clopen sets
$X^\beta=\{\mathcal U:X\in\mathcal U\}$.  The subspace $\om^*$ consists of
free ultrafilters, and $X^*=X^\beta\cap\om^*$.  These spaces are compact
Hausdorff and zero-dimensional.  For completeness, ultrafilters can be
viewed as the Boolean-valued homomorphisms
$\mathcal P(\om)\to\{0,1\}$; their defining equations give a closed
subspace of the compact product $2^{\mathcal P(\om)}$.  The equations
excluding singletons define the closed subspace $\om^*$.  The clopen
coordinate conditions separate points.  Compactness implies that every
clopen set is represented by a finite Boolean combination of coordinates,
hence by a single subset of $\om$.

For a free filter $\mathcal F$, let
\[
 \Ext(\mathcal F)=\{\mathcal U\in\om^*:\mathcal F\subseteq\mathcal U\}.
\]
It is nonempty and compact, and
\begin{equation}\label{eq:filter-intersection}
 \mathcal F=\bigcap\Ext(\mathcal F).
\end{equation}
Indeed, if $X\notin\mathcal F$, adjoining $X^c$ to $\mathcal F$ gives a
proper filter, which extends to an ultrafilter omitting $X$.

\begin{lemma}\label{lem:compact-P}
If every ultrafilter extending $\mathcal F$ is a $P$-point, then
$\Ext(\mathcal F)$ is finite.
\end{lemma}
\begin{proof}
For a $P$-point $\mathcal U$, any sequence of basic neighborhoods
$X_n^*$ of $\mathcal U$ contains a common basic neighborhood $Y^*$.
Choose $Y\in\mathcal U$ with $Y\st X_n$ for all $n$.  Therefore, in the
compact space $K=\Ext(\mathcal F)$, any countable intersection of open sets
is open.  Every countable subset $D$ of such a $T_1$ space is closed, since
its complement is the intersection of the open complements of its
singletons.  It is also discrete, since each $d\in D$ has the open
neighborhood $K\setminus(D\setminus\{d\})$.  If $K$ were infinite, it
would contain a countably infinite $D$, which would be a compact infinite
discrete space.  The cover of $D$ by its singletons contradicts compactness.
\end{proof}

Define the canonical ideal
\[
 \mathcal C_k(\mathcal F)
 =\Id\bigl(\Gfc^{(k)}\cup\{A_X^{(k)}:X\in\mathcal F\}\bigr).
\]
Its members are precisely the hypergraphs contained in
$A_X^{(k)}\cup K$, with $X\in\mathcal F$ and $K\in\Gfc^{(k)}$.

\begin{lemma}\label{lem:core-filter}
The ideals $\mathcal C_k(\mathcal F)$ and
$\mathcal C_k(\mathcal F)+\Lcal_k$ are $W$-ray-proper for every $W$
satisfying \eqref{eq:W}.  Both have canonical filter $\mathcal F$.
\end{lemma}
\begin{proof}
It suffices to treat the larger ideal.  A member is contained in
$A_X^{(k)}\cup K\cup L$, where $X\in\mathcal F$, $K$ has a finite weak
coloring, and $L\in\Lcal_k$.  Some color class meets $X$ infinitely.
By Lemma~\ref{lem:LF-rows}, this intersection contains an infinite
$L$-independent set.  It is independent for the whole union.

The canonical filter contains $\mathcal F$.  Suppose
$A_Y^{(k)}\subseteq A_X^{(k)}\cup K\cup L$.  If $X\setminus Y$ were infinite,
choose an infinite $L$-independent $B\subseteq X\setminus Y$.  Then
$[B]^k\subseteq K$, contradicting the finite weak chromatic number of $K$.
Thus $X\st Y$, whence $Y\in\mathcal F$.
\end{proof}

\begin{theorem}[Canonical ideal classification]\label{thm:core}
For a free filter $\mathcal F$ and $2\le k<\om$, the following are equivalent:
\begin{enumerate}
\item $\mathcal C_k(\mathcal F)+\Lcal_k$ is a $\V_k$-$P$-ideal;
\item $\mathcal C_k(\mathcal F)$ is a $\V_k$-$P$-ideal;
\item for every sequence $c_n:\om\to q_n$, with each $q_n$ a positive
integer, there are $X\in\mathcal F$ and a finite partition
$X=X_0\mathbin{\dot\cup}\cdots\mathbin{\dot\cup}X_{m-1}$ such that
$c_n\restr X_i$ is eventually constant for all $n,i$;
\item every ultrafilter extending $\mathcal F$ is a $P$-point;
\item $\Ext(\mathcal F)$ is a nonempty finite set of $P$-points;
\item $\mathcal F$ is the intersection of a nonempty finite family of
$P$-points.
\end{enumerate}
Eventual constancy on a finite cell in \textup{(iii)} is understood
vacuously.
\end{theorem}
\begin{proof}
We first prove the equivalence of (i)--(iii), using a finite coloring to
partition a set in the filter into cells.  We then relate (iii) to
ultrafilter extensions.

\emph{(i)$\Rightarrow$(iii).}
For the given maps, put
\[
 D_n=\{e\in E_k:c_n\text{ is nonconstant on }e\}.
\]
Then $c_n$ weakly colors $D_n$.  Choose a common relative upper bound
$H\subseteq A_X^{(k)}\cup K\cup L$, with $X\in\mathcal F$,
$K\in\Gfc^{(k)}$, $L\in\Lcal_k$, and finite $F_n$ covering
$D_n\setminus H$.  A finite weak coloring of $K$ partitions $X$ into
independent cells $X_i$.  Inside a cell,
\[
 D_n\cap[X_i]^k\subseteq L\cup\Star_k(F_n).
\]
Delete $F_n$ and a finite exceptional set for local finiteness of $L$.
If two $c_n$-values still occurred infinitely often, fix a vertex with the
first value and choose arbitrarily many $(k-1)$-subsets of the second
infinite color class.  This gives infinitely many hyperedges of $L$
incident with the fixed vertex, a contradiction.  Since $c_n$ has finite
range, precisely one value occurs on a cofinite subset of each infinite
cell.

\emph{(iii)$\Rightarrow$(ii).}
First we show that $\mathcal F$ is a $P$-filter.  For a decreasing sequence
$X_n\in\mathcal F$, apply (iii) to $\mathbf1_{X_n}$.  Obtain $X\in\mathcal F$
and finitely many cells deciding all $X_n$.  Call a cell bad if it is almost
contained in $X_n^c$ for some $n$; this includes any finite cell.  Let $B$ be
the union of the bad cells.  If there are bad cells, choose $N$ at least as
large as all their witnessing indices.  Then $B\cap X_N$ is finite, so
$X\setminus B\in\mathcal F$.  If there are none, put $B=\varnothing$.
Every remaining cell is almost contained in every $X_n$, hence
$X\setminus B$ is the required pseudointersection.

Now take $G_n\subseteq A_{X_n}^{(k)}\cup H_n$ in
$\mathcal C_k(\mathcal F)$, and choose $X\in\mathcal F$ with $X\st X_n$ for
all $n$.  Apply (iii) to finite weak colorings $c_n$ of the $H_n$.  This gives
$Y\in\mathcal F$ and cells $Y_i$ on which all $c_n$ are eventually constant.
Set $Z=X\cap Y$.  Let $K$ consist of the $k$-subsets of $Y$ that meet at
least two cells.  Color each $Y_i$ separately and give $\om\setminus Y$ an
extra color.  Thus $K\in\Gfc^{(k)}$ and
$G=A_Z^{(k)}\cup K\in\mathcal C_k(\mathcal F)$.
The difference $A_{X_n}^{(k)}\setminus A_Z^{(k)}$ is covered by the finite
set $Z\setminus X_n$.  Hyperedges of $H_n$ outside $A_Z^{(k)}\cup K$ lie
inside one cell.  After deleting the finitely many exceptions to the
constant $c_n$-value in each of the finitely many cells, no such edge
remains.  Hence $G_n\setminus G\in\V_k$.

\emph{(ii)$\Rightarrow$(i).}
By Lemma~\ref{lem:LF-P}, $\Lcal_k$ is a $\V_k$-$P$-ideal.
Given a sequence in the sum, bound its canonical parts and its locally
finite parts separately.  Their union is a relative upper bound in the
sum.  Properness has already been proved in Lemma~\ref{lem:core-filter}.

\emph{(iii)$\Rightarrow$(iv).}
Let $\mathcal U\supseteq\mathcal F$ and let $X_n\in\mathcal U$.
Apply (iii) to their characteristic functions.  The resulting set
$X\in\mathcal F$ belongs to $\mathcal U$, so exactly one cell $C$ of the
finite partition belongs to $\mathcal U$.  It is infinite.  Since both
$C$ and $X_n$ lie in $\mathcal U$, the alternative $C\st X_n^c$ is
impossible.  Thus $C\st X_n$ for every $n$.

\emph{(iv)$\Rightarrow$(v)$\Rightarrow$(vi).}
Apply Lemma~\ref{lem:compact-P} and then
\eqref{eq:filter-intersection}.

\emph{(vi)$\Rightarrow$(iii).}
Write $\mathcal F=\bigcap_{i<m}\mathcal U_i$, with $m\ge1$ and all
$\mathcal U_i$ $P$-points.  For each $i,n$, one $c_n$-color class belongs to
$\mathcal U_i$.  Take a pseudointersection $Y_i\in\mathcal U_i$ of those
classes.  Then $Y=\bigcup_{i<m}Y_i$ belongs to every $\mathcal U_i$, hence
to $\mathcal F$.  Disjointify the sets $Y_i$ by deleting from each one the
preceding sets.  Each resulting cell is contained in its original $Y_i$,
so every $c_n$ is eventually constant on it.
\end{proof}

For an ultrafilter $\mathcal U$, Proposition~\ref{prop:trace} gives
\begin{equation}\label{eq:core-trace-ultra}
 \mathcal C_k(\mathcal U)=T_k(\mathcal U).
\end{equation}
Thus $\mathcal C_k(\mathcal U)$ has the relative $P$-property exactly when
$\mathcal U$ is a $P$-point.

\section{Locally finite extensions and capture ideals}\label{sec:capture}

Adding $\Lcal_k$ to a trace ideal preserves its canonical filter and its
relative $P$-property.  The row bounds for $\Lcal_k$ then give explicit
bases for the induced ideals on increasing sequences and determine their
uniformities.

\subsection{Trace ideals with locally finite restrictions}

The relative upper bounds from Lemma~\ref{lem:LF-P} combine with
pseudointersections in the filter to give upper bounds in the sum ideal.

\begin{definition}
For a free filter $\mathcal F$, let
\[
 \mathcal J_{\mathcal F,k}=T_k(\mathcal F)+\Lcal_k.
\]
\end{definition}

\begin{theorem}[Locally finite extensions]\label{thm:extension}
For every free filter $\mathcal F$,
\begin{equation}\label{eq:J-intrinsic}
 H\in\mathcal J_{\mathcal F,k}
 \quad\Longleftrightarrow\quad
 (\exists X\in\mathcal F)\ H\cap[X]^k\in\Lcal_k.
\end{equation}
The ideal $\mathcal J_{\mathcal F,k}$ is $W$-ray-proper for every
$W$ satisfying \eqref{eq:W}, its canonical filter is $\mathcal F$, and
\begin{equation}\label{eq:J-P}
 \mathcal J_{\mathcal F,k}\text{ is a }\V_k\text{-}P\text{-ideal}
 \quad\Longleftrightarrow\quad
 \mathcal F\text{ is a }P\text{-filter}.
\end{equation}
Also, $\Gfc^{(k)}\subseteq\mathcal J_{\mathcal F,k}$ holds exactly when
$\mathcal F$ is an ultrafilter.
\end{theorem}
\begin{proof}
If $H\subseteq G\cup L$ with $G\in T_k(\mathcal F)$ and $L\in\Lcal_k$,
choose $X\in\mathcal F$ with $G\cap[X]^k\in\V_k$.  Then
$H\cap[X]^k\in\Lcal_k$.  Conversely, under the right-hand side of
\eqref{eq:J-intrinsic},
\[
 H\subseteq A_X^{(k)}\cup(H\cap[X]^k)
\]
is a decomposition in the sum.
Within the witnessing infinite $X$, delete finitely many exceptional
vertices and choose an infinite independent set by
Lemma~\ref{lem:LF-rows}.  This proves $W$-ray-properness.
If $A_Y^{(k)}\cap[X]^k\in\Lcal_k$, an infinite $X\setminus Y$ would yield
an infinite complete subhypergraph of a hypergraph locally finite after
finite deletion.  Hence $X\st Y$, proving that the canonical filter is
$\mathcal F$.

If $\mathcal F$ is a $P$-filter, both summands have the relative $P$-property
by Proposition~\ref{prop:trace} and Lemma~\ref{lem:LF-P}; bounding each
part of a sequence separately proves the same property for their proper
sum.  Conversely, take $X_n\in\mathcal F$ and a common relative upper
bound $H\in\mathcal J_{\mathcal F,k}$ for the $A_{X_n}^{(k)}$.  Choose
$Y\in\mathcal F$ with $H\cap[Y]^k\in\Lcal_k$.  If $Y\setminus X_n$ were
infinite, deleting a finite cover of $A_{X_n}^{(k)}\setminus H$ and the
finitely many local-finiteness exceptions would leave an infinite set
$B\subseteq Y\setminus X_n$ with $[B]^k\subseteq H$.  This contradicts local
finiteness.  Thus $Y\st X_n$ for all $n$, proving \eqref{eq:J-P}.

An ultrafilter already has $\Gfc^{(k)}\subseteq T_k(\mathcal F)$.
Conversely, if $\Delta_kX\in\mathcal J_{\mathcal F,k}$, take a witnessing
$Y\in\mathcal F$ in \eqref{eq:J-intrinsic}.  If both $Y\cap X$ and
$Y\setminus X$ were infinite, fixing one vertex in one side and varying
$k-1$ vertices in the other would contradict local finiteness after finite
deletion.  Thus $Y$ almost decides $X$, and $\mathcal F$ is an ultrafilter.
\end{proof}

\subsection{Explicit bases and countable closure}

For a $W$-ray-proper ideal $\mathcal J$, set
\[
 \Icap_W(\mathcal J)
   =\{\mathcal A\subseteq\R:(\exists H\in\mathcal J)
                                    \ \mathcal A\subseteq\Gamma_W(H)\}.
\]
It is a proper ideal by Lemma~\ref{lem:null-ideal}.  If $\mathcal J$ has the
relative $P$-property, then it is a $\sigma$-ideal.  Indeed,
$H_n\setminus H\in\V_k$ implies
$\bigcup_n\Gamma_W(H_n)\subseteq\Gamma_W(H)$.

\begin{theorem}[Capture bases and witness invariance]\label{thm:basis}
Let $\mathcal F$ be a free filter.  For every $W$ satisfying
\eqref{eq:W}, the ideal $\Icap_W(\mathcal J_{\mathcal F,k})$ is the same
ideal $\Icap_k(\mathcal F)$, with cofinal family
\begin{equation}\label{eq:N-basis}
\begin{split}
 N_{X,f}^{(k)}={}&\{R:R\not\st X\}\\
 &{}\cup\{R\in\R:r_{n+k-1}\le f(r_n)
                                     \text{ for infinitely many }n\},
\end{split}
\end{equation}
where $X\in\mathcal F$ and $f\in\mathcal M_k$.
Every set in this cofinal family is $G_\delta$, and
\begin{equation}\label{eq:capture-sigma}
 \Icap_k(\mathcal F)\text{ is a }\sigma\text{-ideal}
 \quad\Longleftrightarrow\quad
 \mathcal F\text{ is a }P\text{-filter}.
\end{equation}
\end{theorem}
\begin{proof}
By \eqref{eq:J-intrinsic} and Lemma~\ref{lem:Bf}, every
$H\in\mathcal J_{\mathcal F,k}$ is contained modulo $\V_k$ in
$A_X^{(k)}\cup B_f^{(k)}$ for some $X\in\mathcal F$ and $f\in\mathcal M_k$.
Conversely, every such union belongs to $\mathcal J_{\mathcal F,k}$.
Lemma~\ref{lem:cuts} identifies $\Gamma_W(A_X^{(k)})$ with
$\{R:R\not\st X\}$.
We also have
\begin{equation}\label{eq:W-bound-equality}
 \Gamma_W(B_f^{(k)})=\Gamma_k(B_f^{(k)}).
\end{equation}
For the nontrivial inclusion, infinitely many hyperedges of
$W(R)\cap B_f^{(k)}$ have infinitely many different minimum vertices,
because the witness is locally finite.  If one minimum is $r_n$, then
$r_{n+k-1}\le\max e\le f(r_n)$, so the consecutive block starting at $r_n$
belongs to $B_f^{(k)}$.  The opposite inclusion follows from
$P_R^{(k)}\subseteq W(R)$.  The finite-union identity for capture sets now
gives \eqref{eq:N-basis} and witness invariance.

Both alternatives in \eqref{eq:N-basis} are infinitely-often conditions on
clopen cylinders, and a finite union of $G_\delta$ sets is $G_\delta$.
The ideal contains all singletons.  For a fixed $R$, the locally finite
hypergraph $P_R^{(k)}$ captures $R$.

If $\mathcal F$ is a $P$-filter, Theorem~\ref{thm:extension} and the preceding
observation give countable closure.  Conversely, assume countable closure
and take $X_n\in\mathcal F$.  Each set $\{R:R\not\st X_n\}$ belongs to the
capture ideal, so their union is captured by one
$H\in\mathcal J_{\mathcal F,k}$.  Choose $Y\in\mathcal F$ with
$H\cap[Y]^k\in\Lcal_k$.  If $Y\setminus X_n$ were infinite, it would
contain an infinite independent set $R$ for $H$.  This $R$ belongs to the
union but is not captured by $H$, a contradiction.  Hence $Y\st X_n$ for
all $n$.
\end{proof}

\subsection{Uniformities and small extensions}

For a $W$-ray-proper ideal, define
\[
 \nu_W(\mathcal J)
 =\min\{|\mathcal A|:\mathcal A\subseteq\R,
             \ (\forall H\in\mathcal J)\ \mathcal A\cap D_H^W\ne\varnothing\}.
\]
If every countable subfamily of $\mathcal J$ has a common avoiding ray,
define
\begin{equation*}
 \nu_{W,\om}(\mathcal J)
 =\min\bigl\{|\mathcal A|:\mathcal A\subseteq\R,\
   (\forall (H_n)_{n<\om}\in\mathcal J^\om)\
   \mathcal A\cap\bigcap_nD_{H_n}^W\ne\varnothing\bigr\}.
\end{equation*}
For consecutive witnesses use $\nu_k$ and $\nu_{k,\om}$.
By definition,
\[
 \nu_W(\mathcal J)=\non(\Icap_W(\mathcal J)).
\]

\begin{proposition}[Trace uniformities]\label{prop:trace-cardinals}
For every free filter $\mathcal F$ and $W$ satisfying \eqref{eq:W},
\[
 \nu_W(T_k(\mathcal F))=\pc(\mathcal F),
 \qquad
 \nu_{W,\om}(T_k(\mathcal F))=\pcs(\mathcal F).
\]
\end{proposition}
\begin{proof}
The hypergraphs $A_X^{(k)}$, $X\in\mathcal F$, are cofinal in
$T_k(\mathcal F)$, and their avoidance sets are exactly
$\{R:R\st X\}$.  Thus the first assertion is the definition of a $\pi$-base.
For a sequence of trace hypergraphs choose corresponding sets $X_n$ and
apply a $\pi_\sigma$-base.  Conversely, the sequence $A_{X_n}^{(k)}$ forces
a common almost subset of all $X_n$.
\end{proof}

\begin{theorem}[Uniformities of locally finite extensions]\label{thm:cardinals}
For every free filter $\mathcal F$, every $k\ge2$, and every $W$ satisfying
\eqref{eq:W},
\begin{align}
 \nu_W(\mathcal J_{\mathcal F,k})
   &=\non(\Icap_k(\mathcal F))
     =\max\{\pc(\mathcal F),\mathfrak d\},\label{eq:single-cardinal}\\
 \nu_{W,\om}(\mathcal J_{\mathcal F,k})
   &=\max\{\pcs(\mathcal F),\mathfrak d\}.\label{eq:countable-cardinal}
\end{align}
If $\mathcal F$ is a $P$-filter, these two values are equal.
\end{theorem}
\begin{proof}
The lower bounds by the two $\pi$-characters follow from
Proposition~\ref{prop:trace-cardinals}.  For the lower bound by $\mathfrak d$,
let $\mathcal A$ avoid every member of $\Lcal_k$.
For $R\in\mathcal A$, let $t_R(m)$ be the $k$th element of
$R\cap[m,\infty)$.  Given a nondecreasing $f\in\mathcal M_k$, choose
$R\in\mathcal A\cap D_{B_f^{(k)}}^W$.
Writing $R=\{r_n:n<\om\}$, we have eventually
$r_{n+k-1}>f(r_n)$.  If $r_n$ is the first element of $R$ at least $m$, then
\[
 t_R(m)=r_{n+k-1}>f(r_n)\ge f(m)
\]
for all sufficiently large $m$.  Thus $\{t_R:R\in\mathcal A\}$ is a
dominating family, and $|\mathcal A|\ge\mathfrak d$.

For the upper bound in \eqref{eq:single-cardinal}, choose a $\pi$-base
$\mathcal B$ of size $\pc(\mathcal F)$ and a dominating family
$\mathcal D$ of size $\mathfrak d$, consisting of increasing functions
$h$ with $h(i)>i$.  Let $e_B:\om\to B$ be the increasing enumeration.
For each pair $(B,h)$, define
\[
 i_0=0,\quad i_{n+1}=h(i_n),\qquad
 R_{B,h}=\{e_B(i_n):n<\om\}.
\]
Fix $G\in\mathcal J_{\mathcal F,k}$, and choose $X\in\mathcal F$ with
$G\cap[X]^k\in\Lcal_k$.  Select $B\in\mathcal B$ with $B\st X$.
The pullback of $G\cap[B]^k$ under $e_B$ has finite rows above some index.
Choose $g(i)$ strictly greater than every index occurring in its $i$th row
for all sufficiently large $i$.  Take $h\in\mathcal D$ with $g\le^*h$.
For sufficiently large $n$, the next selected index $i_{n+1}$ exceeds every
vertex index in the row with minimum $i_n$.  Consequently the tail of
$R_{B,h}$ is independent for $G$.  Hyperedges of $W(R_{B,h})$ using earlier
vertices form a finite set by local finiteness.  Thus $R_{B,h}\in D_G^W$.
The resulting family has size at most
$\pc(\mathcal F)\cdot\mathfrak d=\max\{\pc(\mathcal F),\mathfrak d\}$.

For \eqref{eq:countable-cardinal}, use a $\pi_\sigma$-base $\mathcal B$.
Given $G_n\in\mathcal J_{\mathcal F,k}$, choose $X_n\in\mathcal F$ with
$G_n\cap[X_n]^k\in\Lcal_k$, and one $B\in\mathcal B$ with $B\st X_n$
for every $n$.  Pull back all the induced hypergraphs on $B$.
Lemma~\ref{lem:LF-P} gives a locally finite relative upper bound $L$ for
them.  Choose $g$ bounding the rows of $L$, then $h\in\mathcal D$ as above.
The resulting ray avoids $L$.  Each difference between a pullback and $L$
has a finite vertex cover, so this one ray avoids all the $G_n$.  The lower
bound by $\pcs(\mathcal F)$ uses a countable sequence of endpoint
hypergraphs, and the lower bound by $\mathfrak d$ already follows from the
one-hypergraph case.  Finally use \eqref{eq:pi-P}.
\end{proof}

\begin{theorem}[Small locally finite perturbations]\label{thm:small-extension}
Suppose $\mathcal F$ is a free $P$-filter and
$\mathcal H\subseteq\Lcal_k$ has size less than $\mathfrak b$.  There is a
locally finite hypergraph $L$ such that, for every $W$ satisfying
\eqref{eq:W}, the ideal
\[
 \mathcal K=\Id\bigl(T_k(\mathcal F)\cup\{L\}\bigr)
\]
contains $T_k(\mathcal F)\cup\mathcal H$, is $W$-ray-proper and a
$\V_k$-$P$-ideal, and satisfies
\[
 \nu_W(\mathcal K)=\nu_{W,\om}(\mathcal K)=\pc(\mathcal F).
\]
For any $\pi$-base $\mathcal B$, infinite subsets $B'\subseteq B$, one for
each $B\in\mathcal B$, can be chosen so that $\{B':B\in\mathcal B\}$
avoids every member of $\mathcal K$.
\end{theorem}
\begin{proof}
For $H\in\mathcal H$, choose $f_H\in\mathcal M_k$ from Lemma~\ref{lem:Bf}.
Since $|\mathcal H|<\mathfrak b$, there is $f\in\mathcal M_k$ eventually
dominating all $f_H$.  Set $L=B_f^{(k)}$.  Then
$H\setminus L\in\V_k\subseteq T_k(\mathcal F)$, so $\mathcal K$ contains
$\mathcal H$.  In the countable case one can equivalently use the explicit
row-tail bound \eqref{eq:row-tail-LF}.
For each $B\in\mathcal B$, take an infinite $L$-independent $B'\subseteq B$.
If $G\subseteq A_X^{(k)}\cup L$ with $X\in\mathcal F$, choose $B\st X$.
Then $B'\st X$, and $W(B')$ meets $A_X^{(k)}$ only finitely while missing
$L$.  Thus the refined family avoids $G$.

For a sequence $G_n\subseteq K_n\cup L$ with $K_n\in T_k(\mathcal F)$,
choose a relative upper bound $K\in T_k(\mathcal F)$ by
Proposition~\ref{prop:trace}.  Then $K\cup L$ bounds the sequence in
$\mathcal K$.  Properness follows from the avoiding family.  The lower
bound $\pc(\mathcal F)$ follows from the endpoint hypergraphs.  A family
avoiding every single member of a relative $P$-ideal also avoids every
countable sequence, by applying it to a common relative upper bound.
\end{proof}

Starting with a $\pi$-base of minimum size in
Theorem~\ref{thm:small-extension} gives an avoiding family of size
$\pc(\mathcal F)$.  Adjoining fewer than $\mathfrak b$ members of
$\Lcal_k$ therefore preserves this value.  Adjoining all of $\Lcal_k$
gives the value $\max\{\pc(\mathcal F),\mathfrak d\}$ by
Theorem~\ref{thm:cardinals}.

\section{The Louveau ideal and the arity hierarchy}\label{sec:louveau}

We now specialize the filter to a free ultrafilter $\mathcal U$.  The
$P$-point property determines countable closure of the capture ideals.
The $Q$-point property determines whether their dependence on the arity is
strict.

\subsection{Absorption of locally finite hypergraphs}

An independent set belonging to $\mathcal U$ places a hypergraph in
$T_k(\mathcal U)$.  The $Q$-point property provides such sets for every
locally finite hypergraph by selecting well-separated vertices.

\begin{theorem}[$Q$-points and local finiteness]\label{thm:Q-absorption}
For a free ultrafilter $\mathcal U$ and $k\geq2$, the following are equivalent:
\begin{enumerate}
\item $\mathcal U$ is a $Q$-point;
\item $\Lcal_k\subseteq T_k(\mathcal U)$;
\item $\mathcal J_{\mathcal U,k}=T_k(\mathcal U)$.
\end{enumerate}
\end{theorem}
\begin{proof}
The equivalence of (ii) and (iii) follows from the definition of the ideal
sum.  Assume (ii), and let $\om=\dot\bigcup_n C_n$ be a partition into
finite sets.  The hypergraph
\[
 H=\bigcup_n[C_n]^k
\]
is locally finite.  Choose $X\in\mathcal U$ and a finite vertex cover $F$
of $H\cap[X]^k$.  Then
\[
 |(X\setminus F)\cap C_n|\leq k-1\qquad(n<\om).
\]
Partition $X\setminus F$ into $k-1$ sets, each meeting every $C_n$ in at
most one point.  Since their union belongs to $\mathcal U$, one of these
sets belongs to $\mathcal U$.  Thus $\mathcal U$ is a $Q$-point.

Conversely, assume (i) and let $H\in\Lcal_k$.  Remove a finite vertex set
$F$ so that the remaining hypergraph is locally finite.  Its \emph{two-shadow}
is the graph joining two vertices whenever they occur together in some
hyperedge.  Each vertex belongs to finitely many hyperedges, each of finite
size, so the two-shadow is locally finite.

Choose $0=a_0<a_1<a_2<\cdots$ with $a_1>\max F$ and, recursively,
with every neighbor of a vertex below $a_{n+1}$ lying below $a_{n+2}$.
Local finiteness makes each choice possible.  Put $I_n=[a_n,a_{n+1})$.
An edge of the two-shadow joins vertices in the same interval or in
adjacent intervals.  The $Q$-point property gives $X\in\mathcal U$ with
$|X\cap I_n|\leq1$ for all $n$.  Exactly one of the unions of the even and
the odd intervals belongs to $\mathcal U$.  Intersect $X$ with that union
and remove the finite exceptional vertices, obtaining $Y\in\mathcal U$.
No edge of the two-shadow has both endpoints in $Y$.  Therefore
$H\cap[Y]^k=\varnothing$, and $H\in T_k(\mathcal U)$.
\end{proof}

Petrenko and Protasov give a related characterization of $Q$-points by
thin sets in locally finite metric spaces \cite[Remark~6.2]{PP2015}.

\subsection{A successor-set basis for the Louveau ideal}

Let $\om^{\uparrow<\om}$ be the set of finite strictly increasing sequences
of natural numbers.  We identify a member of $\R$ with its increasing
enumeration.  If $s$ is a nonempty finite sequence, $\max s$ denotes its
last entry; set $\max\varnothing=-1$.  For $i>\max s$, write $s^\frown i$
for the sequence obtained by appending $i$.

A \emph{$\mathcal U$-Laver tree} is a nonempty tree
$T\subseteq\om^{\uparrow<\om}$, closed under initial segments, with a
stem $s_T\in T$ such that every node of $T$ is comparable with $s_T$
under extension, and
\[
 \succT_T(t)=\{i>\max t:t^\frown i\in T\}\in\mathcal U
 \qquad(t\in T,\ s_T\subseteq t).
\]
Its branch set $[T]$ consists of the infinite increasing sequences all of
whose initial segments belong to $T$.  A subtree in this definition is
again a $\mathcal U$-Laver tree, whose stem extends or equals the original stem.

\begin{definition}
The \emph{Louveau ideal} $\louv_{\mathcal U}$ consists of all
$A\subseteq\R$ such that every $\mathcal U$-Laver tree $T$ has a
$\mathcal U$-Laver subtree $S\subseteq T$ with $[S]\cap A=\varnothing$.
\end{definition}

We use the increasing-sequence presentation of the Louveau ideal from
\cite[Section~3, p.~2653]{BS1999}, following Louveau's construction
\cite{Louveau1976}.  Successor sets give an explicit basis for this ideal.

For a family $\mathbf B=(B_s:s\in\om^{\uparrow<\om})$ with
$B_s\in\mathcal U$, define
\begin{equation}\label{eq:successor-null}
 N(\mathbf B)=
 \{R:\text{for infinitely many }n,\ r_n\notin B_{R\restr n}\}.
\end{equation}
We may always replace $B_s$ by $B_s\setminus[0,\max s]$.

\begin{lemma}[Successor-set basis]\label{lem:louveau-basis}
The sets $N(\mathbf B)$ form a cofinal family in $\louv_{\mathcal U}$
under inclusion.  In particular, $\louv_{\mathcal U}$ is a proper
$\sigma$-ideal.
\end{lemma}
\begin{proof}
Given $T$ and $\mathbf B$, retain the stem of $T$ and restrict the successor
set at each later node $s$ to $\succT_T(s)\cap B_s$.  These intersections
belong to $\mathcal U$.  Every branch of the resulting subtree obeys
$B_s$ at all stages beyond its stem, so it misses $N(\mathbf B)$.
Thus every set in \eqref{eq:successor-null} belongs to
$\louv_{\mathcal U}$.

Conversely, let $A\in\louv_{\mathcal U}$.  Call a finite increasing
sequence $s$ \emph{good} if there is a $\mathcal U$-Laver tree with stem
$s$ whose branch set misses $A$.  Put
$G_s=\{i>\max s:s^\frown i\text{ is good}\}$.
If $G_s\in\mathcal U$, the corresponding trees can be joined at $s$ to
form such a tree with stem $s$.  Hence $s$ is good.

If some $s$ were not good, then
$\{i>\max s:i\notin G_s\}\in\mathcal U$.  Repeating this observation at every nongood
node produces a $\mathcal U$-Laver tree with stem $s$ all of whose nodes
above the stem are nongood.  The definition of $\louv_{\mathcal U}$ gives
a subtree with branch set disjoint from $A$.  Its stem is then a good
node of the constructed tree, a contradiction.  Thus every $s$ is good.

For each $s$, choose a witnessing tree $T_s$ with stem $s$.  For a finite
increasing sequence $t$, let
\[
 B_t=(\max t,\infty)\cap
 \bigcap\{\succT_{T_s}(t):s\subseteq t,\ t\in T_s\}.
\]
There are only finitely many initial segments $s$ of $t$, and the displayed
intersection includes $s=t$.  Each factor belongs to $\mathcal U$, so
$B_t\in\mathcal U$.  If $R$ eventually obeys these successor sets, choose
$N$ after its last violation and put $s=R\restr N$.  An induction shows
that $R\restr n\in T_s$ for every $n\geq N$: at such a node, the definition
of $B_{R\restr n}$ includes its successor set in $T_s$.  Hence
$R\in[T_s]$, and $R\notin A$.  This proves $A\subseteq N(\mathbf B)$.

For countable closure, suppose $A_j\subseteq N(\mathbf B^{\,j})$.
At a node $s$, put
\[
 C_s=\bigcap_{j\leq|s|}B_s^{\,j}\in\mathcal U.
\]
A branch eventually obeying $\mathbf C$ eventually obeys each
$\mathbf B^{\,j}$.  Therefore $\bigcup_jA_j\subseteq N(\mathbf C)$.
The whole space is not null, since every $\mathcal U$-Laver tree has an
infinite branch.  A singleton $\{R\}$ is contained in $N(\mathbf B)$ by
setting $B_{R\restr n}=\om\setminus\{r_n\}$ and choosing arbitrary
members of $\mathcal U$ at the remaining nodes.  Downward closure is immediate.
\end{proof}

The cofinal basis in Lemma~\ref{lem:louveau-basis} is the one used by
Brendle and Shelah in their calculation of the cardinal characteristics of
$\louv_{\mathcal U}$ \cite[proof of Theorem~2, p.~2654]{BS1999}.

\subsection{The graph representation of the Louveau ideal}

For a $P$-point, one set $X\in\mathcal U$ is almost contained in every
successor set of a given basis element.  A function $f$ bounds the finite
exceptions at nodes with bounded maximum.  For rays eventually contained
in $X$, the graph bound $r_{n+1}>f(r_n)$ then ensures eventual obedience
to all successor conditions along the ray.

\begin{theorem}[The Louveau representation]\label{thm:louveau-identification}
For every free ultrafilter $\mathcal U$,
\[
 \Icap_2(\mathcal U)\subseteq\louv_{\mathcal U}.
\]
Moreover,
\begin{equation}\label{eq:louveau-identification}
 \Icap_2(\mathcal U)=\louv_{\mathcal U}
 \quad\Longleftrightarrow\quad
 \mathcal U\text{ is a }P\text{-point}.
\end{equation}
\end{theorem}
\begin{proof}
Fix $X\in\mathcal U$ and $f\in\mathcal M_2$.  In any
$\mathcal U$-Laver tree, restrict each successor set above a node $s$ to
$X\cap(f(\max s),\infty)$, using $X$ at the empty node.  Every branch of
the resulting tree is eventually contained in $X$ and eventually satisfies
$r_{n+1}>f(r_n)$.  Thus $N_{X,f}^{(2)}\in\louv_{\mathcal U}$.
The basis theorem for $\Icap_2(\mathcal U)$ gives the inclusion.

Suppose now that $\mathcal U$ is a $P$-point, and let
$A\in\louv_{\mathcal U}$.  By Lemma~\ref{lem:louveau-basis}, choose
$B_s\in\mathcal U$ with $A\subseteq N(\mathbf B)$.  There are countably
many finite increasing sequences, so take $X\in\mathcal U$ such that
$X\st B_s$ for every $s$.  For each $m$, only finitely many increasing
sequences have maximum at most $m$.  Choose a nondecreasing
$f\in\mathcal M_2$ satisfying
\begin{equation}\label{eq:louveau-threshold}
 f(m)>\max\left(\bigcup_{\max s\leq m}(X\setminus B_s)\right)
 \qquad(m<\om),
\end{equation}
with the maximum of the empty set interpreted as $-1$.

If $R\notin N_{X,f}^{(2)}$, then eventually $r_n\in X$ and
$r_{n+1}>f(r_n)$.  Apply \eqref{eq:louveau-threshold} to
$s=R\restr(n+1)$, whose maximum is $r_n$.  It follows that
$r_{n+1}\in B_s$ for all sufficiently large $n$.  Hence
$R\notin N(\mathbf B)$, and therefore $R\notin A$.  We have shown
$A\subseteq N_{X,f}^{(2)}$, so $A\in\Icap_2(\mathcal U)$.

Conversely, equality with $\louv_{\mathcal U}$ makes
$\Icap_2(\mathcal U)$ a $\sigma$-ideal.  By
Theorem~\ref{thm:basis}, this implies that $\mathcal U$ is a $P$-point.
\end{proof}

For comparison, Brendle and Shelah proved
\begin{equation}\label{eq:BS-uniformity}
 \non(\louv_{\mathcal U})
 =\max\{\pcs(\mathcal U),\mathfrak d\}
\end{equation}
for every free ultrafilter \cite[Theorem~2(c), p.~2654]{BS1999}.
For $P$-points they obtained the formula with $\pc(\mathcal U)$
\cite[Corollary~3.2(c), p.~2656]{BS1999}.
Theorems~\ref{thm:cardinals} and \ref{thm:louveau-identification} supply a
hypergraph representation and a direct row-bound proof of that formula in
the $P$-point case.  Higher arities distinguish ideals even when their
uniformities agree.

\subsection{Strict dependence on the arity}

Increasing the arity makes the condition
$r_{n+k-1}\le f(r_n)$ stronger.  To prove strict inclusion when
$\mathcal U$ is not a $Q$-point, we select successive groups of exactly
$k$ vertices from finite partition cells.  Every $(k+1)$-element
hyperedge then meets at least two groups, which the local-finiteness construction
keeps independent of one another.

\begin{theorem}[Arity classification]\label{thm:arity}
For every free ultrafilter $\mathcal U$ and every $k\geq2$,
\begin{equation}\label{eq:arity-inclusion}
 \Icap_{k+1}(\mathcal U)\subseteq\Icap_k(\mathcal U).
\end{equation}
Equality holds in \eqref{eq:arity-inclusion} if and only if $\mathcal U$
is a $Q$-point.  If $\mathcal U$ is not a $Q$-point, there is a closed
set $\mathcal A_k\subseteq\R$ such that
\[
 \mathcal A_k\in\Icap_k(\mathcal U)
                    \setminus\Icap_{k+1}(\mathcal U).
\]
\end{theorem}
\begin{proof}
Take $X\in\mathcal U$ and $f\in\mathcal M_{k+1}$.  The implication
\[
 r_{n+k}\leq f(r_n)\quad\Longrightarrow\quad
 r_{n+k-1}\leq f(r_n)
\]
shows that $N_{X,f}^{(k+1)}\subseteq N_{X,f}^{(k)}$.
This proves the inclusion of the ideals.

If $\mathcal U$ is a $Q$-point, Theorem~\ref{thm:Q-absorption} gives
$\mathcal J_{\mathcal U,k}=T_k(\mathcal U)$ for every $k$.  Every member of
$T_k(\mathcal U)$ is contained in $A_X^{(k)}$ for some $X\in\mathcal U$.
Consequently
\begin{equation}\label{eq:Q-capture}
 \Icap_k(\mathcal U)
 =\{\mathcal A:(\exists X\in\mathcal U)(\forall R\in\mathcal A)
                    \ R\not\st X\},
\end{equation}
which does not depend on $k$.

Suppose that $\mathcal U$ is not a $Q$-point.  Fix a partition
$\om=\dot\bigcup_m C_m$ into finite sets having no selector in
$\mathcal U$.  For every $X\in\mathcal U$ and every positive integer $j$,
there are infinitely many $m$ with
\begin{equation}\label{eq:large-blocks}
 |X\cap C_m|\geq j.
\end{equation}
Indeed, if $j=1$, the assertion follows because a finite union of finite
blocks is finite.  For $j\geq2$, failure of \eqref{eq:large-blocks} would,
after deleting finitely many blocks, bound all intersection sizes by
$j-1$.  Partitioning the remaining $\mathcal U$-set into $j-1$ selectors
would put one selector in $\mathcal U$, a contradiction.

Fix $k\geq2$, and put $H_k=\bigcup_m[C_m]^k$.  Define
\begin{equation}\label{eq:closed-separator}
\begin{split}
 \mathcal A_k=\{R=\{r_n:n<\om\}:\ & (\forall j<\om)(\exists m<\om)\\
 &\{r_{kj},\ldots,r_{kj+k-1}\}\subseteq C_m\}.
\end{split}
\end{equation}
For a fixed $j$, the condition in \eqref{eq:closed-separator} depends only
on the first $k(j+1)$ entries of $R$.  It therefore defines a clopen subset
of $\R$.  Their countable intersection $\mathcal A_k$ is closed in $\R$.
Every $R\in\mathcal A_k$ has infinitely many consecutive $k$-windows in
$H_k$, namely those starting at the indices $kj$.  Thus
\[
 \mathcal A_k\subseteq\Gamma_k(H_k)\in\Icap_k(\mathcal U).
\]
We prove that $\mathcal A_k\notin\Icap_{k+1}(\mathcal U)$.

Let $H\in\mathcal J_{\mathcal U,k+1}$.  Choose $X\in\mathcal U$ such that
$H\cap[X]^{k+1}$ is locally finite after a finite vertex deletion, and
make that deletion from $X$.  We recursively select $k$-element sets
\[
 b_i\subseteq X\cap C_{m_i},\qquad
 \max b_i<\min b_{i+1},
\]
so that no new vertex belongs to an $H$-hyperedge meeting a previously
chosen vertex.  At a given stage the chosen vertex set is finite.  Its
incident hyperedges in $H\cap[X]^{k+1}$, and all vertices of those
hyperedges, form a finite set.  Exclude this set, all vertices below the
previous maximum, and the previously used blocks.  Only finitely many
blocks are affected.  Equation~\eqref{eq:large-blocks} supplies a remaining
block containing at least $k$ points of $X$, from which to select the next
$b_i$.

Let $R=\bigcup_i b_i$.  Since each $b_i$ has exactly $k$ elements and
precedes $b_{i+1}$, it occupies the positions $ki,\ldots,ki+k-1$ in the
increasing enumeration of $R$.  Each $b_i$ lies in one block $C_{m_i}$,
so $R\in\mathcal A_k$.
Every $(k+1)$-element subset of $R$ meets at least two of the clusters
$b_i$.  If it were an $H$-hyperedge, a vertex in its latest cluster would
belong to an $H$-hyperedge meeting an earlier chosen vertex, contrary to the
construction.  Thus
\[
 [R]^{k+1}\cap H=\varnothing.
\]
In particular, $R\notin\Gamma_{k+1}(H)$.  Since $H$ was arbitrary, no
single member of $\mathcal J_{\mathcal U,k+1}$ captures $\mathcal A_k$.
The closed set $\mathcal A_k$ therefore witnesses strict inclusion in
\eqref{eq:arity-inclusion}.
\end{proof}

\begin{corollary}\label{cor:strict-sigma-chain}
If $\mathcal U$ is a $P$-point which is not a $Q$-point, then
\[
 \louv_{\mathcal U}=\Icap_2(\mathcal U)
 \supsetneq\Icap_3(\mathcal U)
 \supsetneq\Icap_4(\mathcal U)\supsetneq\cdots
\]
is a strictly decreasing sequence of proper $\sigma$-ideals, with each
strict inclusion witnessed by a closed subset of $\R$.  Every term has
a cofinal family of $G_\delta$ sets and uniformity
$\max\{\pc(\mathcal U),\mathfrak d\}$.
\end{corollary}
\begin{proof}
Apply Theorems~\ref{thm:basis}, \ref{thm:cardinals},
\ref{thm:louveau-identification}, and \ref{thm:arity}.
\end{proof}

\begin{corollary}\label{cor:higher-louveau}
For every free ultrafilter $\mathcal U$ and every $k\geq3$,
\[
 \Icap_k(\mathcal U)=\louv_{\mathcal U}
 \quad\Longleftrightarrow\quad
 \mathcal U\text{ is both a }P\text{-point and a }Q\text{-point}.
\]
\end{corollary}
\begin{proof}
Equality with the Louveau ideal implies that $\Icap_k(\mathcal U)$ is a
$\sigma$-ideal, so Theorem~\ref{thm:basis} makes $\mathcal U$ a $P$-point.
If it were not a $Q$-point, Theorem~\ref{thm:arity} would give
$\Icap_k(\mathcal U)\subsetneq\Icap_2(\mathcal U)=\louv_{\mathcal U}$.
Conversely, a $P$-point which is a $Q$-point satisfies the required equality
by \eqref{eq:Q-capture} and Theorem~\ref{thm:louveau-identification}.
\end{proof}

\section{Stone duality and endpoint spectra}\label{sec:stone}

The quotient by finite vertex covers translates relative pseudounions into
countable neighborhood properties.  We develop this correspondence for
all finite arities.  Throughout this section, the canonical witnesses are
$P_R^{(k)}$.

\subsection{The quotient space and its endpoint maps}

Let
\[
 \mathbb B_k=\mathcal P(E_k)/\V_k,
 \qquad X_k=\Ult(\mathbb B_k).
\]
Here $\Ult(\mathbb B_k)$ is the Stone space of Boolean ultrafilters on
$\mathbb B_k$.  For $H\subseteq E_k$, its basic clopen set is
\[
 \wh H=\{p\in X_k:[H]_{\V_k}\in p\}.
\]
Equivalently, points of $X_k$ are ultrafilters on $E_k$ which are disjoint
from $\V_k$, and we write $H\in p$ in this representation.
The ultrafilter extension argument shows that these clopen sets separate
points and that every cover by basic clopen sets has a finite subcover.
Thus $X_k$ is compact, Hausdorff, and zero-dimensional.
We use the usual Stone correspondence between Boolean algebras and their
clopen algebras; see \cite{Koppelberg1989}.

The passage to capture sets can identify distinct Boolean classes.
For $k=2$, let $M=\{\{2n,2n+1\}:n<\om\}$.  Then
$M\notin\V_2$, but
\[
 \Gamma_2(E_2)=\Gamma_2(E_2\setminus M)=\R.
\]
Indeed, two consecutive edges of an increasing ray cannot both belong to
the matching $M$, so every ray has infinitely many edges in $E_2\setminus M$.

For later use, Boolean order gives
\begin{equation}\label{eq:stone-order}
 H\setminus K\in\V_k
 \quad\Longleftrightarrow\quad
 \wh H\subseteq\wh K.
\end{equation}
To see the reverse implication directly, if $H\setminus K\notin\V_k$,
its nonzero class extends to an ultrafilter, giving a point in
$\wh H\setminus\wh K$.

For $e=\{i_0<\cdots<i_{k-1}\}$, put $s_j(e)=i_j$.
The inverse image of a finite subset of $\om$ under $s_j$ has a finite
vertex cover.  Hence inverse image induces a Boolean homomorphism
$\mathcal P(\om)/\Fin\to\mathbb B_k$, and a continuous map
\[
 \pi_j:X_k\longrightarrow\om^*,
 \qquad X\in\pi_j(p)\ \Longleftrightarrow\ s_j^{-1}[X]\in p.
\]
Define the closed equalizer
\[
 Z_k=\{p\in X_k:\pi_0(p)=\cdots=\pi_{k-1}(p)\}.
\]
On $Z_k$ write $\pi$ for the common endpoint map.

\begin{proposition}\label{prop:equalizer}
For $p\in X_k$,
\[
 p\in Z_k
 \quad\Longleftrightarrow\quad
 \Delta_kX\notin p\text{ for every }X\subseteq\om
 \quad\Longleftrightarrow\quad
 p\cap\Gfc^{(k)}=\varnothing.
\]
\end{proposition}
\begin{proof}
For a fixed $X$, the set $\Delta_kX$ is the finite union of the symmetric
differences $s_i^{-1}[X]\triangle s_j^{-1}[X]$, $i<j<k$.
An ultrafilter avoids this union exactly when all the sets
$s_j^{-1}[X]$ have the same membership value in it.  Requiring this for
every $X$ is precisely equality of the endpoint ultrafilters.
Every cut has finite weak chromatic number.  Conversely, a finite weak
coloring can be encoded by finitely many binary coordinates, so every
member of $\Gfc^{(k)}$ is contained in a finite union of cuts by
Lemma~\ref{lem:cuts}.  Primeness of $p$ completes the proof.
\end{proof}

For a proper ideal $\mathcal J\supseteq\V_k$, set
\begin{equation}\label{eq:Stone-kernel}
 L_{\mathcal J}=\{p\in X_k:p\cap\mathcal J=\varnothing\}
 =\bigcap_{H\in\mathcal J}(X_k\setminus\wh H).
\end{equation}
This is a nonempty closed set.  Conversely, a nonempty closed
$L\subseteq X_k$ defines
\[
 \mathcal J_L=\{H\subseteq E_k:\wh H\cap L=\varnothing\}.
\]
These operations are mutually inverse and reverse inclusion.  For example,
if $H\notin\mathcal J$, its class is nonzero in
$\mathcal P(E_k)/\mathcal J$; an ultrafilter on that quotient supplies
$p\in L_{\mathcal J}\cap\wh H$.  This proves
$\mathcal J_{L_{\mathcal J}}=\mathcal J$.  The converse follows by
separating a point outside a closed set with a clopen set.

\subsection{Relative pseudounions and closed sets}

A \emph{$P$-set} in a topological space is a closed set $L$ such that every
countable intersection of neighborhoods of $L$ is again a neighborhood of
$L$.  This definition will always refer to a specified ambient space.

The restriction of $\V_k$ to $P_R^{(k)}$ is the finite ideal.  Finite
vertex covers meet it finitely, and every finite hypergraph has a finite
vertex cover.  Therefore the clopen set $\wh{P_R^{(k)}}\subseteq X_k$ is
homeomorphic to $\om^*$.
A nonempty closed set $L\subseteq X_k$ will be called \emph{ray-positive}
if every clopen neighborhood of $L$ contains some $\wh{P_R^{(k)}}$.

\begin{theorem}[Stone correspondence for relative ideals]
\label{thm:stone}
Let $\mathcal J$ be a proper ideal on $E_k$ containing $\V_k$.  Then
\begin{enumerate}
\item $\Gfc^{(k)}\subseteq\mathcal J$ if and only if
      $L_{\mathcal J}\subseteq Z_k$;
\item $\mathcal J$ is a $\V_k$-$P$-ideal if and only if
      $L_{\mathcal J}$ is a $P$-set in $X_k$;
\item $\mathcal J$ is ray-proper if and only if $L_{\mathcal J}$ is
      ray-positive.
\end{enumerate}
\end{theorem}
\begin{proof}
For (i), the forward direction follows from Proposition~\ref{prop:equalizer}.
If $H\in\Gfc^{(k)}\setminus\mathcal J$, the quotient-ultrafilter argument
following \eqref{eq:Stone-kernel} gives a point of $L_{\mathcal J}$
containing $H$.  It cannot lie in $Z_k$, proving the converse.

Assume that $\mathcal J$ is a relative $P$-ideal.  Given open neighborhoods
$O_n$ of $L_{\mathcal J}$, compactness and zero-dimensionality give clopen
neighborhoods $X_k\setminus\wh{G_n}\subseteq O_n$.  By the correspondence
between ideals and closed sets, $G_n\in\mathcal J$.
Choose $G\in\mathcal J$ with
$G_n\setminus G\in\V_k$.  Equation~\eqref{eq:stone-order} yields
\[
 L_{\mathcal J}\subseteq X_k\setminus\wh G
 \subseteq\bigcap_n(X_k\setminus\wh{G_n})\subseteq\bigcap_nO_n.
\]
Thus the kernel is a $P$-set.
Conversely, for $G_n\in\mathcal J$, apply the $P$-set property to the
clopen neighborhoods $X_k\setminus\wh{G_n}$, then refine their common
neighborhood to a clopen neighborhood $X_k\setminus\wh G$.
Its containment of the kernel gives $G\in\mathcal J$, and its inclusion
in every $X_k\setminus\wh{G_n}$ gives $G_n\setminus G\in\V_k$.

Every clopen neighborhood of the kernel has the form
$X_k\setminus\wh G$ with $G\in\mathcal J$.  It contains
$\wh{P_R^{(k)}}$ exactly when $P_R^{(k)}\cap G\in\V_k$, which is equivalent
to finiteness of that intersection.  This proves (iii).
\end{proof}

For a proper ideal $\mathcal J\supseteq\Gfc^{(k)}$, the \emph{endpoint spectrum} is the
nonempty compact set
\[
 K_{\mathcal J}=\pi[L_{\mathcal J}]\subseteq\om^*.
\]

\begin{proposition}\label{prop:spectrum}
For every proper $\mathcal J\supseteq\Gfc^{(k)}$,
\begin{equation}\label{eq:spectrum-formula}
 \mathcal F_{\mathcal J}=\bigcap K_{\mathcal J},
 \qquad
 K_{\mathcal J}=\Ext(\mathcal F_{\mathcal J}).
\end{equation}
For a free ultrafilter $\mathcal U$,
\begin{equation}\label{eq:full-fiber}
 L_{T_k(\mathcal U)}=Z_k\cap\pi^{-1}(\{\mathcal U\})=:Z_{k,\mathcal U}.
\end{equation}
For a free filter $\mathcal F$,
\[
 L_{\mathcal C_k(\mathcal F)}=Z_k\cap\pi^{-1}[\Ext(\mathcal F)].
\]
\end{proposition}
\begin{proof}
If $p\in Z_k$ and $\pi(p)=\mathcal U$, then
\[
 [X]^k=\bigcap_{j<k}s_j^{-1}[X]\in p
 \quad\Longleftrightarrow\quad X\in\mathcal U.
\]
Taking complements shows that $A_X^{(k)}$ is avoided by every point of
$L_{\mathcal J}$ exactly when $X$ belongs to every member of
$K_{\mathcal J}$.  This is the first equality of
\eqref{eq:spectrum-formula}.

For any nonempty compact $K\subseteq\om^*$, one has
$K=\Ext(\bigcap K)$.  Indeed, an ultrafilter $\mathcal U\notin K$ can be
separated from $K$ by a clopen set $X^*$ containing $K$ and missing
$\mathcal U$.  Then $X\in\bigcap K$ but $X\notin\mathcal U$.
This proves the second equality.

By Proposition~\ref{prop:trace}, every member of $T_k(\mathcal U)$ is
contained in some $A_X^{(k)}$ with $X\in\mathcal U$.  The ideal also
contains $\Gfc^{(k)}$.  Avoiding it is consequently equivalent to lying
in $Z_k$ and having endpoint $\mathcal U$.  This proves
\eqref{eq:full-fiber}.  For $\mathcal C_k(\mathcal F)$, avoiding its
generators means lying in $Z_k$ and having an endpoint containing every
$X\in\mathcal F$.  This gives the last equality.
\end{proof}

\subsection{Reflection from a single endpoint}

\begin{lemma}[Transitivity of $P$-sets]\label{lem:P-transitivity}
Let $X$ be compact Hausdorff, let $L$ be a $P$-set in $X$, and let $C$ be
a $P$-set in the relative topology of $L$.  Then $C$ is a $P$-set in $X$.
\end{lemma}
\begin{proof}
Let $O_n$ be open neighborhoods of $C$ in $X$.  By the relative $P$-set
property, choose an open $V\subseteq X$ with
$C\subseteq L\cap V\subseteq L\cap\bigcap_nO_n$.
By compactness and regularity, choose an open neighborhood $U$ of $C$
whose closure misses $L\setminus V$.  The closed sets
\[
 F_n=\overline U\cap(X\setminus O_n)
\]
are disjoint from $L$.  The $P$-set property of $L$, applied to
$X\setminus F_n$, gives a neighborhood $W$ of $L$ disjoint from
$\bigcup_nF_n$.  Then $U\cap W$ is a neighborhood of $C$ contained in
every $O_n$.
\end{proof}

A point $x$ of a space $K$ is a \emph{relative $P$-point} if $\{x\}$ is a
$P$-set in $K$.  For $\mathcal U\in\om^*$, this is equivalent to the
ultrafilter definition of a $P$-point.  In fact, the basic neighborhoods
are $X^*$, $X\in\mathcal U$, and a basic neighborhood $Y^*$ refines all
$X_n^*$ exactly when $Y\st X_n$ for all $n$.

\begin{theorem}[Endpoint reflection]\label{thm:reflection}
Let $\varnothing\neq C\subseteq Z_k$ be a $P$-set in $X_k$.  If
$\pi[C]=\{\mathcal U\}$, then $\mathcal U$ is a $P$-point.
More generally, if $\mathcal J\supseteq\Gfc^{(k)}$ is a proper
$\V_k$-$P$-ideal and $\mathcal U$ is a relative $P$-point of
$K_{\mathcal J}$, then $\mathcal U$ is a $P$-point of $\om^*$.
\end{theorem}
\begin{proof}
The ideal $\mathcal J_C$ contains $\Gfc^{(k)}$ and is a relative
$P$-ideal by Theorem~\ref{thm:stone}.  Proposition~\ref{prop:spectrum}
gives $\mathcal F_{\mathcal J_C}=\mathcal U$.
Lemma~\ref{lem:canonical-filter} then makes $\mathcal U$ a $P$-filter.
As it is an ultrafilter, it is a $P$-point.

For the second assertion, the continuous surjection
$\pi:L_{\mathcal J}\to K_{\mathcal J}$ is closed.  Given a relative open
neighborhood $O$ of its fiber over $\mathcal U$, the set
$K_{\mathcal J}\setminus\pi[L_{\mathcal J}\setminus O]$ is an open
neighborhood of $\mathcal U$ whose inverse image is contained in $O$.
For countably many such $O$, the relative $P$-point property gives a common
neighborhood of $\mathcal U$.  Hence the fiber is a relative $P$-set in
$L_{\mathcal J}$.  The latter is an ambient $P$-set by
Theorem~\ref{thm:stone}.  Lemma~\ref{lem:P-transitivity} makes the fiber
an ambient $P$-set, and the first assertion applies.
\end{proof}

In particular, $Z_{k,\mathcal U}$ is a $P$-set if and only if
$\mathcal U$ is a $P$-point, by \eqref{eq:full-fiber},
Proposition~\ref{prop:trace}, and Theorem~\ref{thm:stone}.

\subsection{Equal spectra with different transversal numbers}

For a ray-positive closed set $L\subseteq X_k$, define its ray transversal
number by $\nu_k(L)=\nu_k(\mathcal J_L)$.  Thus it is the least size of a
family $\mathcal A\subseteq\R$ such that every clopen neighborhood of
$L$ contains $\wh{P_R^{(k)}}$ for some $R\in\mathcal A$.

\Needspace{13\baselineskip}
\begin{theorem}[Equal spectra with distinct transversal numbers]\label{thm:spectrum-separation}
Suppose $\mathcal U$ is a $P$-point with
$\pc(\mathcal U)<\mathfrak d$.  For every $k\geq2$, the closed sets
\[
 L_0=L_{T_k(\mathcal U)},\qquad
 L_1=L_{\mathcal J_{\mathcal U,k}}
\]
are nonempty ray-positive $P$-sets satisfying
\[
 L_1\subsetneq L_0,\qquad
 \pi[L_0]=\pi[L_1]=\{\mathcal U\},
\]
whereas
\[
 \nu_k(L_0)=\pc(\mathcal U)<\mathfrak d=\nu_k(L_1).
\]
\end{theorem}
\begin{proof}
Proposition~\ref{prop:trace} and Theorem~\ref{thm:extension} show that both
ideals are proper, ray-proper, and relative $P$-ideals, with common
canonical filter $\mathcal U$.  Their spectra are consequently
$\{\mathcal U\}$ by Proposition~\ref{prop:spectrum}.
Their transversal numbers are given by
Proposition~\ref{prop:trace-cardinals} and Theorem~\ref{thm:cardinals}.
The different numbers force strict inclusion of the ideals, and therefore
strict reverse inclusion of their Stone kernels.
\end{proof}

We record how the hypothesis is realized in a familiar model.  The
\emph{ultrafilter number} is
$\mathfrak u=\min\{\chi(\mathcal U):\mathcal U\text{ is a free ultrafilter}\}$,
where $\chi(\mathcal U)$ is the least size of a base contained in
$\mathcal U$.

\begin{lemma}\label{lem:small-character-P}
If a free ultrafilter $\mathcal U$ satisfies
$\chi(\mathcal U)<\mathfrak d$, then it is a $P$-point.
\end{lemma}
\begin{proof}
Suppose that $X_0\supseteq X_1\supseteq\cdots$ belongs to $\mathcal U$
and has no pseudointersection in $\mathcal U$.  We may arrange $X_0=\om$.
The sets $X_n\setminus X_{n+1}$ together with $\bigcap_nX_n$ form a
countable partition into sets outside $\mathcal U$, after empty cells are
omitted.  There must be infinitely many nonempty cells.  Enumerate them
as $C_n$.  No set meeting every $C_n$ finitely belongs to $\mathcal U$:
such a set would be almost contained in every $X_n$.

Take a base $\mathcal B\subseteq\mathcal U$ of size $\chi(\mathcal U)$
for actual inclusion.  A base for almost inclusion can be made one for
actual inclusion by adding its finite deletions, with no change of this
infinite cardinality.  Define
\[
 g_B(n)=\min\left(B\cap\bigcup_{j\geq n}C_j\right)
 \qquad(B\in\mathcal B).
\]
The minimum exists because the tail union belongs to $\mathcal U$.
Given a nondecreasing $f\in\om^\om$, the set
\[
 Y_f=\bigcup_j(C_j\cap[0,f(j)))
\]
meets each cell finitely, so it does not belong to $\mathcal U$.
Choose $B\in\mathcal B$ contained in its complement.  If
$g_B(n)\in C_j$, then $j\geq n$ and
$g_B(n)\geq f(j)\geq f(n)$.  Hence the family $(g_B:B\in\mathcal B)$
is dominating, contradicting $\chi(\mathcal U)<\mathfrak d$.
\end{proof}

\begin{corollary}\label{cor:model}
If $\mathfrak u=\aleph_1<\mathfrak d=\aleph_2$, there is a $P$-point
$\mathcal U$ which is not a $Q$-point and for which, for every $k\geq2$,
\[
 \nu_k(T_k(\mathcal U))=\aleph_1,
 \qquad
 \non(\Icap_k(\mathcal U))=\aleph_2.
\]
For this ultrafilter, the strict $\sigma$-ideal sequence in
Corollary~\ref{cor:strict-sigma-chain} and the spectral separation in
Theorem~\ref{thm:spectrum-separation} both hold.
\end{corollary}
\begin{proof}
Choose $\mathcal U$ with $\chi(\mathcal U)=\aleph_1$.
Lemma~\ref{lem:small-character-P} makes it a $P$-point.
By Section~\ref{sec:prelim}, a free ultrafilter has no countable
$\pi$-base.  Since $\pc(\mathcal U)\le\chi(\mathcal U)=\aleph_1$,
we obtain $\pc(\mathcal U)=\aleph_1$.
If $\mathcal U$ were a $Q$-point, Theorem~\ref{thm:Q-absorption} would
identify $T_k(\mathcal U)$ with $\mathcal J_{\mathcal U,k}$, contradicting
their different transversal numbers.  The other conclusions now follow
from the preceding theorems.
\end{proof}

The hypothesis of Corollary~\ref{cor:model} holds in the Miller model
\cite[p.~326]{RS2020}.

\section{Countable block products}\label{sec:blocks}

On a countable partition into infinite blocks, local relative upper bounds
can be combined by including the first $n+1$ members of a given sequence
on the $n$th block.  The resulting Boolean quotient is a product, and its
Stone space has a natural map to $\beta\om$ given by the block indices.
We use the consecutive witnesses $P_R^{(k)}$ throughout.
Let
\[
 \om=\mathop{\dot\bigcup}_{n<\om}B_n,
 \qquad |B_n|=\aleph_0,
 \qquad E_n=[B_n]^k.
\]
All local definitions use the order inherited from $\om$.
Write $\V_{k,n}$ and $\Gfc^{(k)}(B_n)$ for the local finite-cover and
finite-chromatic ideals.  Given ideals $\mathcal J_n$ on $E_n$, define
\begin{equation}\label{eq:block-product}
 \mathcal J=
 \{H\subseteq E_k:H\cap E_n\in\mathcal J_n\text{ for all }n<\om\}.
\end{equation}
The set of all hyperedges meeting more than one block belongs to
$\mathcal J$; we call these the cross-block hyperedges.

\subsection{The relative $P$-property and the Boolean product}

\begin{theorem}[Countable block products]\label{thm:block-product}
Suppose each $\mathcal J_n$ is a proper ray-proper
$\V_{k,n}$-$P$-ideal containing $\Gfc^{(k)}(B_n)$.
Then the ideal in \eqref{eq:block-product} is proper, ray-proper, contains
$\Gfc^{(k)}$, and is a $\V_k$-$P$-ideal.  Moreover,
\begin{equation}\label{eq:Boolean-product}
 \mathcal P(E_k)/\mathcal J
 \cong\prod_{n<\om}\mathcal P(E_n)/\mathcal J_n.
\end{equation}
\end{theorem}
\begin{proof}
A finite weak coloring restricts to one on each block, proving the stated
containment.  If $H\in\mathcal J$, choose a block $B_n$ and use local
ray-properness to find $R\subseteq B_n$ with
$|P_R^{(k)}\cap H|<\om$.  This proves ray-properness and properness.

Let $G_i\in\mathcal J$ for $i<\om$.  On each block, choose
$H_n\in\mathcal J_n$ such that
\[
 (G_i\cap E_n)\setminus H_n\in\V_{k,n}\qquad(i<\om).
\]
Set
\[
 H'_n=H_n\cup\bigcup_{i\leq n}(G_i\cap E_n)\in\mathcal J_n,
\]
and let $H$ contain all cross-block hyperedges and satisfy
$H\cap E_n=H'_n$.  Then $H\in\mathcal J$.
For a fixed $i$, the difference $G_i\setminus H$ lies in the finitely many
blocks $n<i$.  Its intersection with each of these blocks has a finite
vertex cover, and the union of those finite covers is a global finite
vertex cover.  Thus $G_i\setminus H\in\V_k$.

The Boolean homomorphism
\[
 H\longmapsto
 \langle[H\cap E_n]_{\mathcal J_n}:n<\om\rangle
\]
is onto.  Choose a representative on each disjoint block and take their
union.  Its kernel is exactly $\mathcal J$, which proves
\eqref{eq:Boolean-product}.
\end{proof}

\subsection{Principal and nonprincipal fibers}

Put $L=L_{\mathcal J}\subseteq Z_k$.  Define the block-index function
$e:\om\to\om$ by $e(i)=n$ for $i\in B_n$.
For an ultrafilter $\mathcal U$ on $\om$, its image under $e$ is
\[
 e(\mathcal U)=\{A\subseteq\om:e^{-1}[A]\in\mathcal U\}\in\beta\om.
\]
This continuous extension is denoted by $\beta e$.
The common endpoint map defines
\begin{equation}\label{eq:rho}
 \rho=\beta e\circ\pi:L\longrightarrow\beta\om.
\end{equation}
For $q\in\beta\om$, let $L_q=\rho^{-1}(\{q\})$.
For $A\subseteq\om$, write
$A^\beta=\{q\in\beta\om:A\in q\}$; principal points are included in this
clopen set.

\begin{theorem}[Block fibers]\label{thm:block-fibers}
Under the hypotheses of Theorem~\ref{thm:block-product}, the map
\eqref{eq:rho} is onto and every fiber $L_q$ is ray-positive.
For a principal point $n$, the fiber $L_n$ is naturally homeomorphic to
$L_{\mathcal J_n}$ in the local Stone space, and $\bigcup_nL_n$ is dense
in $L$.
If $q\in\om^*$ and $p\in L_q$, then $\pi(p)$ is not a $P$-point.
\end{theorem}
\begin{proof}
Choose $p_n\in L_{\mathcal J_n}$ for every $n$.  For $q\in\beta\om$,
define an ultrafilter on $E_k$ by
\[
 H\in p_q
 \quad\Longleftrightarrow\quad
 \{n:H\cap E_n\in p_n\}\in q.
\]
The ultrafilter axioms follow coordinatewise from those of the $p_n$ and
$q$.  The ultrafilter $p_q$ avoids $\mathcal J$, since every member of
$\mathcal J$ is avoided on every coordinate.  Thus $p_q\in L$.
For $A\subseteq\om$, all endpoint preimages of $e^{-1}[A]$ have full
intersection with $E_n$ if $n\in A$, and empty intersection if $n\notin A$.
Consequently $\rho(p_q)=q$, proving surjectivity.
For principal $q=n$, this construction identifies the fiber with the Stone
space of the $n$th quotient in \eqref{eq:Boolean-product}.

More explicitly, $\wh{E_n}\subseteq X_k$ is homeomorphic to the local
Stone space of $\mathcal P(E_n)/\V_{k,n}$.  A point $p\in L$ lies in
$\wh{E_n}$ exactly when all its endpoints contain $B_n$, and hence
exactly when $\rho(p)=n$.  Therefore
\begin{equation}\label{eq:principal-clopen}
 L_n=L\cap\wh{E_n}.
\end{equation}
The product representation shows density of their union.  Indeed, every
nonempty relatively clopen set in $L$ is represented by a nonzero tuple
in \eqref{eq:Boolean-product}; some coordinate is nonzero, and an
ultrafilter on that coordinate gives a point of the clopen set in $L_n$.

Fix $q\in\beta\om$, and let $O=X_k\setminus\wh G$ be a clopen
neighborhood of $L_q$.  The compact set $K=L\cap\wh G$ has closed image
$\rho[K]$ not containing $q$.  Choose a clopen $A^\beta$ with
$q\in A^\beta$ and $A^\beta\cap\rho[K]=\varnothing$.
For every $n\in A$, the principal fiber $L_n$ misses $\wh G$, whence
$G\cap E_n\in\mathcal J_n$ by the local correspondence between ideals
and closed sets.  Choose one such $n$ and a local ray $R\subseteq B_n$
avoiding $G\cap E_n$.  Then $\wh{P_R^{(k)}}\subseteq O$.
This proves ray-positivity of every fiber.

Finally, suppose $q\in\om^*$, $p\in L_q$, and $\pi(p)=\mathcal U$.
For every $m$,
\[
 Y_m=\om\setminus\bigcup_{n\leq m}B_n\in\mathcal U.
\]
If $\mathcal U$ were a $P$-point, choose $A\in\mathcal U$ with
$A\st Y_m$ for all $m$.  Then $A\cap B_n$ is finite for every $n$.
Hence $[A]^k\cap E_n=[A\cap B_n]^k\in\mathcal J_n$, so
$[A]^k\in\mathcal J$.
On the other hand, $A\in\pi_j(p)$ for each $j<k$, and therefore
$[A]^k=\bigcap_{j<k}s_j^{-1}[A]\in p$.  This contradicts $p\in L$.
\end{proof}

\subsection{Minimal closed sets and singleton spectra}

\begin{theorem}\label{thm:minimal}
Every ray-positive closed $L\subseteq Z_k$ contains an inclusion-minimal
ray-positive closed subset $C$.  Every such $C$ has
$\pi[C]=\{\mathcal U\}$ for a unique free ultrafilter $\mathcal U$.
\end{theorem}
\begin{proof}
Consider a decreasing chain of nonempty ray-positive closed subsets of
$L$.  Compactness makes its intersection $C$ nonempty.  If $O$ is a clopen
neighborhood of $C$, some member of the chain is contained in $O$.
Otherwise the closed sets obtained by removing $O$ would have the finite
intersection property, yielding a point of $C\setminus O$.
The contained member supplies a ray clopen set in $O$, so $C$ remains
ray-positive.  Zorn's lemma now gives an inclusion-minimal member.

Fix $X\subseteq\om$, and put $Q_X=\wh{[X]^k}$.
Since $C\subseteq Z_k$,
\[
 C=(C\cap Q_X)\mathbin{\dot\cup}(C\cap Q_{X^c}).
\]
At least one piece is ray-positive.  If both failed to be ray-positive,
choose clopen neighborhoods $O_0\subseteq Q_X$ and
$O_1\subseteq Q_{X^c}$ of the respective pieces, neither containing a
ray clopen set.  For an empty piece use the empty neighborhood.
Their union contains $C$, and hence contains $\wh{P_R^{(k)}}$ for some
$R$.  Since
\[
 Q_X\cup Q_{X^c}=X_k\setminus\wh{\Delta_kX},
\]
the ray avoids $\Delta_kX$.  Lemma~\ref{lem:cuts} gives $R\st X$ or
$R\st X^c$.  The entire clopen set $\wh{P_R^{(k)}}$ then lies in one
of $Q_X,Q_{X^c}$ and hence in the corresponding $O_i$, a contradiction.

Minimality implies $C\subseteq Q_X$ or $C\subseteq Q_{X^c}$ for every
$X$.  The choices determine an ultrafilter
$\mathcal U=\{X:C\subseteq Q_X\}$; finite intersections are preserved,
and finite sets have empty $Q_X$.  Each point of $C$ has common endpoint
$\mathcal U$, proving the assertion and uniqueness.
\end{proof}

\begin{corollary}\label{cor:minimal-nonP}
If a ray-positive $P$-set $L_0\subseteq Z_k$ exists, then there is a
ray-positive $P$-set $L\subseteq Z_k$ containing a minimal ray-positive
closed set whose endpoint is not a $P$-point.
\end{corollary}
\begin{proof}
Take the ideal dual to $L_0$, copy it onto each infinite block, and apply
Theorem~\ref{thm:block-product}.  The resulting kernel $L$ is a
ray-positive $P$-set by Theorem~\ref{thm:stone}.
Choose $q\in\om^*$.  By Theorem~\ref{thm:block-fibers}, $L_q$ is
ray-positive and every endpoint in it is non-$P$.
Theorem~\ref{thm:minimal} supplies a minimal ray-positive closed
$C\subseteq L_q$ with one endpoint.  That endpoint is non-$P$.
\end{proof}

\begin{theorem}[Localization of relative $P$-sets]\label{thm:localize-P}
Let $L$ be the block-product kernel.  Every nonempty relative $P$-set
$M\subseteq L$ meets some principal fiber $L_n$, and $M\cap L_n$ is a
nonempty relative $P$-set in $L$.  In particular, $\rho^{-1}[\om^*]$
contains no nonempty relative $P$-set of $L$.
\end{theorem}
\begin{proof}
If $M$ missed all $L_n$, each $L\setminus L_n$ would be a clopen
neighborhood of $M$.  Their intersection would contain a nonempty
relatively open neighborhood of $M$, disjoint from the dense set
$\bigcup_nL_n$, a contradiction.  Since $L_n$ is relatively clopen,
$M\cap L_n$ is again a relative $P$-set.
\end{proof}

\begin{theorem}[Singleton spectra in block products]\label{thm:selective-product}
The block-product kernel $L$ contains a nonempty $P$-set $C$ of $X_k$
with $|\pi[C]|=1$ if and only if some local kernel $L_{\mathcal J_n}$
contains a nonempty $P$-set of its local Stone space with a single endpoint.
\end{theorem}
\begin{proof}
The local Stone space embeds as the clopen set $\wh{E_n}\subseteq X_k$.
A $P$-set of that clopen space is a $P$-set of $X_k$, so a local witness
gives a global one.
Conversely, a global witness $C\subseteq L$ is a relative $P$-set in $L$.
By Theorem~\ref{thm:localize-P}, $C\cap L_n$ is nonempty for some $n$.
Equation~\eqref{eq:principal-clopen} gives
$C\cap L_n=C\cap\wh{E_n}$, a clopen part of the ambient $P$-set $C$.
It remains an ambient $P$-set and has the same unique endpoint.
Under the local homeomorphism it gives the required witness.
\end{proof}

\subsection{The reduced block product}

Define
\[
 \mathcal J_{\rm red}
 =\{H\subseteq E_k:H\cap E_n\in\mathcal J_n
                         \text{ for all but finitely many }n\}.
\]
It contains $\Gfc^{(k)}$ and is ray-proper, since any member can be
avoided in a block where its restriction belongs to $\mathcal J_n$.
Allowing finitely many exceptional blocks destroys the relative
$P$-property.

\begin{proposition}\label{prop:reduced-product}
Under the hypotheses of Theorem~\ref{thm:block-product},
$\mathcal J_{\rm red}$ is not a $\V_k$-$P$-ideal.
More generally, a ray-proper $\V_k$-$P$-ideal cannot contain both every
$E_n$ and all cross-block hyperedges.
\end{proposition}
\begin{proof}
Suppose a ray-proper relative $P$-ideal $\mathcal K$ contains these sets.
Take a common relative upper bound $H\in\mathcal K$ of the $E_n$.
Choose finite $F_n\subseteq B_n$ covering $E_n\setminus H$, so that
\[
 [B_n\setminus F_n]^k\subseteq H.
\]
The union $H'$ of $H$ and all cross-block hyperedges belongs to
$\mathcal K$.  Let $R=\{r_i:i<\om\}$ be any increasing sequence.
If the block indices of consecutive vertices change infinitely often,
infinitely many consecutive $k$-windows meet more than one block, and hence
belong to $H'$.  If the indices change only finitely often, the sequence
is eventually contained in some $B_n$.  Its late windows then avoid the
finite set $F_n$ and belong to $H$.  Thus every ray meets $H'$ infinitely,
contradicting ray-properness.
The reduced product contains all the stipulated sets, so the general
assertion applies to it.
\end{proof}

\section{Countable decision and extension problems}\label{sec:reaping}

The preceding classification results also give an exact formulation of
several extension questions motivated by reaping.  We first specify the
cardinal parameters and then give the row criterion for adjoining a
pseudounion while preserving a fixed avoiding family.

\subsection{The finite and countable reaping numbers}

A set $X\subseteq\om$ \emph{splits} an infinite $R\subseteq\om$ if both
$R\cap X$ and $R\setminus X$ are infinite.  A family
$\mathcal A\subseteq\R$ is \emph{reaping} if no single $X$ splits all its
members.  The least size of a reaping family is $\mathfrak r$.
A family is \emph{$\sigma$-reaping} if for every sequence
$X_n\subseteq\om$ some $R\in\mathcal A$ decides all the $X_n$, where
deciding means $R\st X_n$ or $R\st X_n^c$.
Its least possible size is $\mathfrak r_\sigma$.
Vojt\'a\v{s}'s question asks whether these two cardinals are equal in ZFC
\cite{Vojtas1992,Blass2010}.

\begin{lemma}\label{lem:finite-reaping}
There is a family of size $\mathfrak r$ which simultaneously decides every
finite family of subsets of $\om$.
\end{lemma}
\begin{proof}
Fix a reaping family $\mathcal A$ of size $\mathfrak r$.
Set $\mathcal A_1=\mathcal A$.  Given $\mathcal A_m$, choose a bijection
$e_B:\om\to B$ for each $B\in\mathcal A_m$ and put
\[
 \mathcal A_{m+1}=\{e_B[C]:B\in\mathcal A_m,\ C\in\mathcal A\}.
\]
Given $X_0,\ldots,X_m$, first choose $B\in\mathcal A_m$ deciding the first
$m$ sets.  Then choose $C\in\mathcal A$ deciding the inverse image of
$X_m\cap B$ under $e_B$.  The subset $e_B[C]\subseteq B$ decides all
$m+1$ sets.  Every $\mathcal A_m$ has size at most $\mathfrak r$, and
$\bigcup_{m\geq1}\mathcal A_m$ has size $\mathfrak r$ and the desired
property.  We used that $\mathfrak r$ is uncountable, which follows by
splitting any countable family of infinite sets with a recursive choice
of two fresh points from each set at infinitely many stages.
\end{proof}

\begin{proposition}\label{prop:reaping-representation}
For every $k\geq2$,
\[
 \nu_k(\Gfc^{(k)})=\mathfrak r,
 \qquad
 \nu_{k,\om}(\Gfc^{(k)})=\mathfrak r_\sigma.
\]
\end{proposition}
\begin{proof}
An avoiding family applied to the cuts $\Delta_kX$ is a reaping family by
Lemma~\ref{lem:cuts}, proving the first lower bound.  Conversely, take the
family from Lemma~\ref{lem:finite-reaping}.  A finite weak coloring of a
hypergraph gives finitely many color classes; an infinite set deciding
all these classes is almost contained in one of them.  Its sufficiently
late consecutive windows therefore avoid the hypergraph.

For a countable family of finite-chromatic hypergraphs, the color classes
of fixed finite colorings form a countable family.  A $\sigma$-reaping
witness deciding all those classes avoids every hypergraph.  The converse
again uses the countable sequence of cuts associated with a given
sequence $X_n$.
\end{proof}

Let $D_X=\{R:R\st X\text{ or }R\st X^c\}$.
Write $\Fray$ for the filter on $\R$ generated by all $D_X$, and write
$\Frays$ for the family of supersets of sets of the form
$\bigcap_nD_{X_n}$.

\begin{proposition}\label{prop:decision-filter}
The family $\Frays$ is the least countably complete proper filter
containing $\Fray$, and
\[
 \tr(\Fray)=\mathfrak r,
 \qquad \tr(\Frays)=\mathfrak r_\sigma.
\]
For every $k\geq2$,
\[
 \{H\subseteq E_k:D_H\in\Frays\}=\Gfc^{(k)}.
\]
\end{proposition}
\begin{proof}
For any sequence $X_n$, recursively choose decreasing infinite sets
$C_n$ deciding the first $n$ sets by taking an infinite side of each
successive partition.  A diagonal increasing choice $r_n\in C_n$ gives
$R\in\bigcap_nD_{X_n}$.  Thus the basic sets are nonempty.
Countable intersections of basic sets are again basic by enumerating the
sets in a double sequence.  This proves properness and countable
completeness.  Any countably complete filter containing every $D_X$
contains these basic intersections, proving minimality.
The transversal formulas follow from Lemma~\ref{lem:finite-reaping}
and the definition of $\sigma$-reaping.  The final equality is
Theorem~\ref{thm:absolute}.
\end{proof}

For the refining relation on $\R$
\[
 X\mathrel{\mathrm{ref}}R
 \quad\Longleftrightarrow\quad R\st X\text{ or }R\st X^c,
\]
the ideal dual to $\Frays$ is the ideal of families which are
not $\mathrm{ref}$-$\om$-dominating, in the terminology of
\cite[Definitions~2.9 and~7.6]{Schilhan2026}.  Explicitly, it consists of
all families contained in a countable union of splitting sets
$\{R:X\text{ splits }R\}$.

If $\mathcal J\supseteq\Gfc^{(k)}$ is ray-proper and a relative
$P$-ideal, then
\begin{equation}\label{eq:envelope-bound}
 \mathfrak r_\sigma\leq\nu_k(\mathcal J).
\end{equation}
Indeed, for any sequence of cuts, take a common relative upper bound
$H\in\mathcal J$.  A ray avoiding $H$ decides every set defining those
cuts.  Thus an avoiding family for $\mathcal J$ is $\sigma$-reaping, proving
\eqref{eq:envelope-bound} directly.

\subsection{Row bounds preserving an avoiding family}

The row-tail bounds and the functions $\lambda_R$ are defined in
Section~\ref{subsec:row-bounds}.  We apply their normal form to preservation
of a fixed avoiding family.

Let $\mathcal A\subseteq\R$ avoid every member of a ray-proper ideal
$\mathcal J$.  The sets $\mathcal A\cap D_K$, $K\in\mathcal J$, form a
base of a proper filter $\mathscr F_{\mathcal J,\mathcal A}$ on
$\mathcal A$.  A subset of $\mathcal A$ is \emph{positive} for this filter
if it meets every one of its members.

\begin{proposition}[One-step preservation]\label{prop:one-step}
Let $G_0\subseteq G_1\subseteq\cdots$ belong to $\mathcal J$.
There is a common $\V_k$-upper bound $H$ such that $\mathcal A$ avoids
every member of $\Id(\mathcal J\cup\{H\})$ if and only if some strictly
increasing $h\in\om^\om$ makes
\[
 \{R\in\mathcal A\cap\bigcap_nD_{G_n}:\lambda_R\leq^*h\}
\]
positive for $\mathscr F_{\mathcal J,\mathcal A}$.
\end{proposition}
\begin{proof}
The family $\mathcal A$ avoids every member of
$\Id(\mathcal J\cup\{H\})$ exactly when
\[
 \mathcal A\cap D_K\cap D_H\neq\varnothing\qquad(K\in\mathcal J).
\]
Thus the condition is positivity of $\mathcal A\cap D_H$.
If an upper bound $H$ has this property, Lemma~\ref{lem:row-normal-form}
gives $H_h\subseteq H$, so $D_H\subseteq D_{H_h}$ and positivity is
preserved.  Equation~\eqref{eq:row-avoidance} gives the displayed set.
Conversely, a positive set of the displayed form permits adjoining $H_h$,
which is a relative upper bound by the same lemma.
\end{proof}

\begin{corollary}\label{cor:small-character-extension}
Suppose $C=\mathcal A\cap\bigcap_nD_{G_n}$ is positive for
$\mathscr F_{\mathcal J,\mathcal A}$, and this filter has a base of size
less than $\mathfrak b$.  Then the one-step extension in
Proposition~\ref{prop:one-step} exists.
\end{corollary}
\begin{proof}
For each member $B$ of a base of size less than $\mathfrak b$, choose
$R_B\in B\cap C$.  The family of functions $\lambda_{R_B}$ has a common
$\leq^*$-upper bound.  Increase that bound to a strictly increasing $h$.
The corresponding set in Proposition~\ref{prop:one-step} meets every
basic set $B$, and is therefore positive.
\end{proof}

\subsection{Further questions}

\begin{question}\label{q:envelope}
Does ZFC prove the existence of a ray-proper $\V_2$-$P$-ideal
$\mathcal J\supseteq\Gfc^{(2)}$ with $\nu_2(\mathcal J)=\mathfrak r$?
\end{question}

By \eqref{eq:envelope-bound}, an affirmative answer gives
$\mathfrak r=\mathfrak r_\sigma$.
If there is a $P$-point $\mathcal U$ with $\pc(\mathcal U)=\mathfrak r$,
then $T_2(\mathcal U)$ already supplies the required ideal.
Theorem~\ref{thm:spectrum-separation} also shows that enlargements with the
same endpoint spectrum can increase the relevant transversal number.

Proposition~\ref{prop:one-step} gives two requirements for an extension
construction.  The set $C=\mathcal A\cap\bigcap_nD_{G_n}$ must be
positive, and for some $h$ the subset
$\{R\in C:\lambda_R\leq^*h\}$ must also be positive.
Theorem~\ref{thm:small-extension} verifies both requirements for small
families of locally finite perturbations of trace ideals, after refining
a minimum-size $\pi$-base without changing its cardinality.

\begin{question}\label{q:selective}
For $k\geq2$, does every ray-positive $P$-set $L$ of $X_k$ with
$L\subseteq Z_k$ contain a
nonempty $P$-set of $X_k$ concentrated over one endpoint?
\end{question}

Theorem~\ref{thm:reflection} makes such an endpoint a $P$-point.
Corollary~\ref{cor:minimal-nonP} shows that minimality of a ray-positive
closed subset does not ensure this conclusion.  Theorem~\ref{thm:selective-product}
reduces the existence question for a countable block product to the
corresponding question in one of its components.

For a $P$-point $\mathcal U$ which is not a $Q$-point, the ideals
$\Icap_k(\mathcal U)$ form a strictly decreasing sequence with equal
uniformities and closed sets separating successive terms.  It remains
to determine their other cardinal characteristics and the reducibility
relations between distinct terms.

\section*{Declaration of AI use}
ChatGPT was used for proofreading and language polishing.

\begin{thebibliography}{99}
\normalsize
\interlinepenalty=10000

\bibitem{Blass2010}
A.~Blass,
\emph{Combinatorial cardinal characteristics of the continuum},
in Handbook of Set Theory (M.~Foreman and A.~Kanamori, eds.),
Springer, Dordrecht, 2010, pp.~395--489.
\newline\url{https://doi.org/10.1007/978-1-4020-5764-9_7}

\bibitem{BS1999}
J.~Brendle and S.~Shelah,
\emph{Ultrafilters on $\omega$---their ideals and their cardinal characteristics},
Trans. Amer. Math. Soc. 351 (1999), 2643--2674.
\newline\url{https://doi.org/10.1090/S0002-9947-99-02257-6}

\bibitem{HMM2010}
M.~Hru\v{s}\'ak, D.~Meza-Alc\'antara, and H.~Minami,
\emph{Pair-splitting, pair-reaping and cardinal invariants of $F_\sigma$-ideals},
J. Symbolic Logic 75 (2010), 661--677.
\newline\url{https://doi.org/10.2178/jsl/1268917498}

\bibitem{Koppelberg1989}
S.~Koppelberg,
\emph{General theory of Boolean algebras},
in Handbook of Boolean Algebras, Vol.~1 (J.~D.~Monk and R.~Bonnet, eds.),
North-Holland, Amsterdam, 1989, pp.~1--307.

\bibitem{Louveau1976}
A.~Louveau,
\emph{Une m\'ethode topologique pour l'\'etude de la propri\'et\'e de Ramsey},
Israel J. Math. 23 (1976), 97--116.
\newline\url{https://doi.org/10.1007/BF02756789}

\bibitem{PP2015}
O.~V.~Petrenko and I.~V.~Protasov,
\emph{Ultrafilters on $G$-spaces},
Algebra Discrete Math. 19 (2015), 254--269.

\bibitem{RS2020}
D.~Raghavan and S.~Shelah,
\emph{A small ultrafilter number at smaller cardinals},
Arch. Math. Logic 59 (2020), 325--334.
\newline\url{https://doi.org/10.1007/s00153-019-00693-8}

\bibitem{Schilhan2026}
J.~Schilhan,
\emph{Universal domination and idealized forcing},
preprint, 2026, \href{https://arxiv.org/abs/2608.18964v1}{arXiv:2608.18964v1}.

\bibitem{Vojtas1992}
P.~Vojt\'a\v{s},
\emph{Cardinalities of noncentered systems of subsets of $\omega$},
Discrete Math. 108 (1992), 125--129.
\newline\url{https://doi.org/10.1016/0012-365X(92)90666-4}

\end{thebibliography}
\end{document}