\documentclass[11pt]{article} \usepackage{iftex} \ifPDFTeX\usepackage[utf8]{inputenc}\fi \usepackage[T1]{fontenc} \usepackage{amsmath,amssymb,amsthm} \usepackage[margin=1.2in]{geometry} \usepackage[dvipsnames]{xcolor} \usepackage[colorlinks,linkcolor=blue,citecolor=blue,urlcolor=black]{hyperref} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{fact}[theorem]{Fact} \newtheorem*{claim}{Claim} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newenvironment{claimproof} {\begin{proof}[Proof of the claim]\renewcommand{\qedsymbol}{$\square_{\mathrm{Claim}}$}} {\end{proof}} \newcommand{\N}{\mathbb{N}} \newcommand{\w}{\omega} \newcommand{\R}{\mathcal{R}} \newcommand{\FIN}{\operatorname{\mathsf{FIN}}} \newcommand{\Exh}{\operatorname{\mathsf{Exh}}} \newcommand{\res}{\restriction} \newcommand{\KG}{\mathrm{KG}} \newcommand{\chif}{\chi_{\mathrm{f}}} \newcommand{\EDfin}{\mathcal{ED}_{\mathrm{fin}}} \newcommand{\hg}{\mathcal{H}} \newcommand{\chromsubG}{\varphi^G_{\chi}} \newcommand{\fracsubG}{\varphi^G_{\chi_f}} \newcommand{\chromsub}{\varphi_{\chi}} \newcommand{\fracsub}{\varphi_{\chi_f}} \newcommand{\hallazgo}[1]{{\color{BrickRed}\textsf{\small[CL/H: #1]}}} \newcommand{\taller}[1]{{\color{Sepia}\textsf{\small[CL/T: #1]}}} \hypersetup{pdftitle={The random graph ideal and fractional coloring}, pdfauthor={Carlos Lopez-Callejas}} \title{The random graph ideal and fractional coloring} \author{Carlos L\'opez-Callejas\thanks{Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel. \texttt{lopezcc@biu.ac.il}. Supported by the Israel Science Foundation (Grant Agreement 203/22).}} \date{} \begin{document} \maketitle \begin{abstract} The random graph ideal is pathological, answering a question of L\'opez-Abad et al. We prove this by constructing a positive hereditarily pathological restriction using Kneser graphs. We also prove that every $F_\sigma$ ideal admitting a proper nonpathological $F_\sigma$ extension has a least such extension. Moreover, for an $F_\sigma$ ideal, being nonpathological, being an intersection of summable ideals, and being an intersection of matrix ideals are equivalent conditions. This answers both converses asked by Filip\'ow and Tryba. The least nonpathological $F_\sigma$ extension of the random graph ideal is proper and is described by the fractional cochromatic numbers of its finite induced subgraphs. \end{abstract} \medskip \noindent\emph{2020 MSC:} Primary 03E15; Secondary 03E05, 05C15, 05C72, 05C63, 05C65. \emph{Keywords:} $F_\sigma$ ideal; pathological ideal; nonpathological submeasure; random graph; fractional coloring; least nonpathological extension; summable ideal; matrix ideal; Kat\v{e}tov order. \section{Introduction}\label{sec:intro} Submeasure representations are a fundamental tool in the study of definable ideals on countable sets. Classical theorems of Mazur \cite{Mazur} and Solecki \cite{Solecki} provide representations by lower semicontinuous submeasures for $F_\sigma$ ideals and analytic $P$-ideals, respectively. These representations lead to the distinction between pathology and nonpathology, studied in particular by Farah \cite{Farah}. A submeasure is nonpathological if it is a supremum of measures; an $F_\sigma$ ideal is nonpathological if it admits such a representation by a nonpathological submeasure, and pathological otherwise. We recall the precise definitions in Section~\ref{sec:prelim}. For analytic $P$-ideals, Hru\v{s}\'ak's measure dichotomy yields a characterization of nonpathology through the Kat\v{e}tov order \cite[Corollary~5.6]{Hrusak}. More recent work has focused on pathology for $F_\sigma$ ideals and its connections with representations in Banach spaces \cite{MMU,LOU}, following earlier work on representations of analytic $P$-ideals \cite{BNFP,BF}. These results motivate the search for combinatorial criteria that distinguish pathological from nonpathological $F_\sigma$ ideals. The random graph ideal $\R$, generated by the cliques and independent sets of the countable random graph, is a natural example in this setting. It is a tall $F_\sigma$ ideal closely connected to Ramsey properties and the Kat\v{e}tov order \cite{HMTU,Hrusak}; see also \cite{CDU} for recent work on Ramsey properties of ideals. A theorem of Hru\v{s}\'ak, Meza-Alc\'antara, Th\"ummel and Uzc\'ategui shows that every tall $F_\sigma$ ideal has a positive restriction above $\R$ in the Kat\v{e}tov order \cite{HMTU}. L\'opez-Abad, Olmos-Prieto and Uzc\'ategui-Aylwin asked whether $\R$ is pathological \cite[Question~5.19]{LOU}. We answer this question affirmatively by constructing a positive hereditarily pathological restriction: every further positive restriction of this witness is pathological. Together with a positive restriction isomorphic to the nonpathological ideal $\EDfin$, this gives the two local behaviors of Corollary~\ref{cor:portrait}. The pathological witness is an induced disjoint union of finite Kneser graphs. Their unbounded chromatic numbers and independent star covers of uniformly positive density satisfy the block criterion of Mart\'inez, Meza-Alc\'antara and Uzc\'ategui \cite[Theorem~3.8]{MMU}, which also applies on every positive restriction of the witness. For the nonpathological restriction, universality gives an induced copy of a disjoint union of finite complete graphs of increasing sizes, whose homogeneous-set ideal is isomorphic to $\EDfin$ \cite[Examples~3.5 and~4.5]{MMU}. Proposition~\ref{prop:solecki} also shows that a positive hereditarily pathological restriction of $\R$ lies above Solecki's ideal in the Kat\v{e}tov order, connecting the Kneser construction with the ideal appearing in Hru\v{s}\'ak's measure dichotomy. We use covering representations to prove in Theorem~\ref{thm:least-extension} that every $F_\sigma$ ideal admitting a proper nonpathological $F_\sigma$ extension has a least such extension. To describe the construction, let $\mathcal I$ be a proper $F_\sigma$ ideal on $V$. Choose a compact hereditary family $\mathcal C\subseteq2^V$ containing the singletons and generating $\mathcal I$; such a family exists for every $F_\sigma$ ideal. Write $\varphi_{\mathcal C}(A)$ for the least number of members of $\mathcal C$ covering $A$, with value $\infty$ when no finite cover exists. The nonpathological envelope $\widehat{\varphi_{\mathcal C}}$ is the supremum of all measures dominated by $\varphi_{\mathcal C}$. Then \[ \mathcal J_{\mathcal I}=\FIN(\widehat{\varphi_{\mathcal C}}) \] is the least nonpathological $F_\sigma$ extension of $\mathcal I$ when a proper such extension exists, and equals $\mathcal P(V)$ otherwise. In particular, $\mathcal J_{\mathcal I}$ depends only on $\mathcal I$, not on the choice of $\mathcal C$. To prove that this extension is the least one, we compare the covering envelope with any nonpathological representation of an extension of $\mathcal I$. If $\widehat{\varphi_{\mathcal C}}(A)<\infty$, classical finite covering duality \cite[Theorem~2.1]{CJ} and compactness give a probability on the generators contained in $A$ such that every vertex of $A$ is covered with a uniformly positive probability. This covering bound shows that every nonpathological lower semicontinuous submeasure that is finite on the generators is also finite on $A$. Filip\'ow and Tryba \cite[Theorem~11.2]{FT} and, independently, Figueroa-Sierra and Hru\v{s}\'ak \cite[Corollary~6.1.2]{FigueroaThesis} proved that every nonpathological $F_\sigma$ ideal is an intersection of summable ideals. Combining this representation with the least-extension theorem, we identify $\mathcal J_{\mathcal I}$ with the intersection of all proper summable extensions of $\mathcal I$. Corollary~\ref{cor:matrix-intersection} also identifies it with the intersection of all matrix ideals containing $\mathcal I$. Consequently, for an $F_\sigma$ ideal, being nonpathological, being an intersection of summable ideals, and being an intersection of matrix ideals are equivalent conditions. This answers both converses in \cite[Question~11.6]{FT}. Corollary~\ref{cor:analytic-p-extension} shows that every nonpathological analytic $P$-ideal containing $\mathcal I$ also contains $\mathcal J_{\mathcal I}$. In particular, every $F_\sigma$ ideal representable by unconditional convergence in a Banach space is also representable by bounded finite subsums, answering \cite[Question~2.4]{LOU}. For a countable graph $H$, the covering submeasure associated with the family of independent sets is $\chromsub(A)=\chi(H[A])$. Define the fractional submeasure by \[ \fracsub(A)=\sup\{\chif(H[F])\mid F\subseteq A\text{ finite}\}. \] Finite covering duality gives $\widehat{\chromsub}=\fracsub$, as we show in Theorem~\ref{thm:env}.\footnote{Here the supremum is taken over finite induced subgraphs: assigning weights to all independent sets of an infinite graph can give a strictly larger value \cite[Theorem~6 and the following remark]{GP}.} If $H$ has no infinite clique, independent sets generate its homogeneous-set ideal $\mathcal I_H$. Assuming that $\mathcal I_H$ is proper, Theorem~\ref{thm:nonpath-char} shows that it is nonpathological if and only if, for every $C>0$, the chromatic numbers $\chi(H[F])$ are bounded as $F$ ranges over the finite vertex sets satisfying $\chif(H[F])\leq C$. For a disjoint union of finite graphs, this bound must hold uniformly over the induced subgraphs of all blocks. The criterion gives partial answers to \cite[Questions~5.21 and~5.22]{LOU} for graphs with no infinite clique. The proof adapts the extraction argument in \cite[proof of Theorem~17]{GIPSSST}. If the condition fails, it produces a positive set whose induced graph is a disjoint union of finite graphs with unbounded chromatic numbers and uniformly bounded fractional chromatic numbers. The restriction to this set is hereditarily pathological. The least-extension theorem also describes super-pathology, the property of admitting no proper nonpathological $F_\sigma$ extension. In Theorem~\ref{thm:restriction-extension}, we prove that $\mathcal J_{\mathcal I}$ consists exactly of the members of $\mathcal I$ together with the positive sets $A$ for which $\mathcal I\res A$ is super-pathological. In particular, every pathological $F_\sigma$ ideal has a positive super-pathological restriction. For a graph $H$ with no infinite clique and $A\in\mathcal I_H^+$, the restriction $\mathcal I_H\res A$ is super-pathological exactly when $\fracsub(A)<\infty$. For $\R$, the general construction uses covers by both cliques and independent sets. Theorem~\ref{thm:B} shows that its least nonpathological $F_\sigma$ extension is proper and is computed by the fractional cochromatic numbers of finite induced subgraphs of $R$. Thus, $\R$ is not super-pathological, whereas its Kneser restriction is. For $d\geq3$, pathology of the random $d$-uniform hypergraph ideal $\R_d$ and the existence of positive hereditarily pathological restrictions already follow from the constructions in \cite[Theorem~5.14 and Corollary~5.15]{LOU}. We recover these conclusions from the graph case: in Theorem~\ref{thm:Rd-copy}, we use a standard extension of graph colorings to obtain a positive restriction of $\R_d$ isomorphic to $\R$ for every $d\geq2$. This also transfers the positive nonpathological restriction isomorphic to $\EDfin$. Comparing the homogeneous-cover submeasures and their envelopes, we show that the least nonpathological $F_\sigma$ extension of $\R_d$ is proper. Finally, for every $d\geq3$ and $n\geq d+1$, the homogeneous-set ideal of the universal $K_n^{(d)}$-free hypergraph is pathological but not super-pathological. Remark~\ref{rem:LOU} explains why the super-pathology assertion in \cite[Corollary~5.18]{LOU} does not follow from the positive restrictions constructed there. Section~\ref{sec:prelim} fixes the notation and recalls the submeasure framework. Section~\ref{sec:local} proves pathology of $\R$ and describes its two local behaviors. Section~\ref{sec:generators} proves the least-extension theorem and studies summable, matrix and nonpathological analytic $P$-ideal extensions. Section~\ref{sec:coloring} proves the chromatic criterion for graphs with no infinite clique and derives the corresponding block criteria. Section~\ref{sec:global} describes the least extension of $\R$, applies Pr\"omel--R\"odl universality to Erd\H{o}s sequences, and relates the envelope to Kelley covering densities. Section~\ref{sec:hypergraphs} transfers the random-graph results to uniform hypergraphs. Section~\ref{sec:super} describes super-pathological restrictions, gives the block classification, and treats universal hypergraphs with a finite complete hypergraph forbidden. \section{Preliminaries}\label{sec:prelim} Let $\w=\{0,1,\ldots\}$, with each natural number identified with the set of its predecessors. For a set $X$ and $k\in\w$, write $[X]^k$ for the family of subsets of $X$ of cardinality $k$. All underlying sets are countably infinite unless stated otherwise. An ideal $\mathcal I$ on $X$ is closed under subsets and finite unions, contains all finite subsets of $X$, and is \emph{proper}, i.e., $X\notin\mathcal I$. For an ideal $\mathcal I$ on $X$, put $\mathcal I^+=\mathcal P(X)\setminus\mathcal I$ and $\mathcal I\res Y=\mathcal I\cap\mathcal P(Y)$. This restriction is proper precisely when $Y\in\mathcal I^+$. We identify $\mathcal P(X)$ with $2^X$ via characteristic functions and equip it with the product topology. An ideal on $X$ is $F_\sigma$ if it is a countable union of closed sets in this topology, and is tall if every infinite subset of $X$ contains an infinite member of the ideal. All notions are invariant under bijections of the underlying sets. A \emph{lower semicontinuous submeasure} (lscsm) on $X$ is a map $\varphi:\mathcal P(X)\to[0,\infty]$ satisfying $\varphi(\emptyset)=0$, monotonicity, subadditivity, $\varphi(\{x\})<\infty$ for $x\in X$, and \[ \varphi(A)=\sup\{\varphi(F)\mid F\subseteq A\text{ finite}\}. \] Write $\FIN(\varphi)=\{A\subseteq X\mid \varphi(A)<\infty\}$. The proper ideals of this form are exactly the $F_\sigma$ ideals, by Mazur's theorem \cite{Mazur}; see also \cite[Theorem~1.5]{Hrusak2011}. A \emph{measure} means a countably additive function $\mu(A)=\sum_{x\in A}m(x)$, where $m:X\to[0,\infty)$ is finite at each point. For submeasures write $\psi\leq\varphi$ for pointwise domination and let $\mathcal M(X,\varphi)$ be the measures dominated by $\varphi$. The submeasure $\varphi$ is \emph{nonpathological} if \[ \varphi(A)=\sup_{\mu\in\mathcal M(X,\varphi)}\mu(A) \quad(A\subseteq X). \] This follows the countably additive convention of \cite[\S3]{MMU}; for lsc submeasures it is equivalent to Farah\textquotesingle s notion \cite[p.~21]{Farah}, by \cite[Proposition~4.1(7a)]{FT}. For $F_\sigma$ ideals we use \cite[Definition~3.1]{MMU}: $\mathcal I$ is \emph{nonpathological} if $\mathcal I=\FIN(\varphi)$ for some nonpathological lscsm, and \emph{pathological} otherwise. The ideal-level definition concerns all representations, not just a chosen submeasure. An $F_\sigma$ ideal is \emph{hereditarily pathological} if $\mathcal I\res Y$ is pathological for every $Y\in\mathcal I^+$. \begin{fact}[Restriction]\label{fact:restr} Restrictions of nonpathological submeasures are nonpathological. Consequently, every positive restriction of a nonpathological $F_\sigma$ ideal is nonpathological. Equivalently, an $F_\sigma$ ideal with a pathological positive restriction is pathological. \end{fact} Indeed, if $\mathcal I=\FIN(\varphi)$ with $\varphi$ nonpathological, then $\mathcal I\res Y=\FIN(\varphi\res\mathcal P(Y))$, and positivity makes this ideal proper. The last assertion follows by contraposition. For a graph $H=(V,E)$ and $A\subseteq V$, write $H[A]=(A,E\cap[A]^2)$ for the subgraph induced by $A$. A set is \emph{homogeneous} if it is a clique or an independent set. Write $\hg(H)$ for this family and $\langle\hg(H)\rangle$ for the sets covered by finitely many members of it. The chromatic number $\chi(H)$ is the least number of independent sets covering $V$, with $\chi(\emptyset)=0$ and value $\infty$ when no finite independent cover exists. A proper coloring with $m$ colors is a map $c:V\to m$ whose fibers are independent. Since every finite independent cover can be refined to a partition, $\chi(H)$ is equivalently the least number of colors in a proper coloring. \section{The random graph ideal: pathology and local extremes}\label{sec:local} The random graph $R=(\w,E)$ is the unique countable graph with the extension property: for disjoint finite $a,b\subseteq\w$, some vertex outside $a\cup b$ is adjacent to every point of $a$ and none of $b$. Every countable graph occurs as an induced subgraph \cite[Proposition~4.1]{Cameron}. Its homogeneous-set ideal $\R=\langle\hg(R)\rangle$ is proper, tall and $F_\sigma$ \cite[p.~834]{Hrusak}. For an induced copy $H=R[X]$, hereditary closure of homogeneity gives \begin{equation}\label{eq:induced} \R\res X=\langle\hg(H)\rangle: \end{equation} one restricts a finite homogeneous cover to $X$, and conversely every homogeneous subset of $H$ is homogeneous in $R$. \begin{lemma}[Block representation]\label{lem:transfer} Let $G=\bigsqcup_{n\in\omega}G_n$ be a disjoint union of nonempty finite graphs, with $V_n=V(G_n)$ and $V=\bigsqcup_nV_n$. For $A\subseteq V$, put \[ \chromsub(A)=\sup_{n\in\omega}\chi(G_n[A\cap V_n]). \] Then $\chromsub$ is an lscsm and $\chromsub(A)=\chi(G[A])$ for every $A\subseteq V$. Moreover, $\chromsub(\{x\})=1$ for every $x\in V$ and $\chromsub(V_n)=\chi(G_n)$ for every $n\in\omega$. If $H$ is a countable graph with $V\subseteq V(H)$ and $H[V]=G$, then \[ \langle\hg(H)\rangle\res V=\langle\hg(G)\rangle=\FIN(\chromsub). \] This is a proper $F_\sigma$ ideal exactly when $\sup_{n\in\omega}\chi(G_n)=\infty$; in that case $V$ is positive for $\langle\hg(H)\rangle$. \end{lemma} \begin{proof} Fix $A\subseteq V$. Restricting independent covers to each block gives $\chromsub(A)\leq\chi(G[A])$. Conversely, if $m=\chromsub(A)<\infty$, for each $n\in\omega$ choose independent subsets $(I_i^n)_{i0$. Then $\FIN(\varphi)$ is pathological. \end{lemma} \begin{remark}[The multiplicity adaptation]\label{rem:multiset} The paragraph before \cite[Theorem~3.8]{MMU} explicitly permits arbitrary finite partitions. Our only change is to allow repeated terms. The measure estimate in \cite[Lemma~3.7]{MMU} still holds, since counting each occurrence separately gives \[ \sum_{i0. \] If $R[X]\cong\bigsqcup_{n\in\w}\KG(m_n,k_n)$, then $X\in\R^+$ and $\R\res X$ is hereditarily pathological. \end{theorem} \begin{proof} We apply Lemma~\ref{lem:blocks} on every positive subset of $X$. Write $X=\bigsqcup_{n\in\w}X_n$ for the blocks of the induced copy, and let $S_i^n\subseteq X_n$ be the image of the star $S_i(m_n,k_n)$. Put \[ \varphi(A)=\sup_{n\in\w}\chi(R[A\cap X_n]) \qquad(A\subseteq X). \] By Lemma~\ref{lem:transfer}, $\varphi$ is an lscsm and $\R\res X=\FIN(\varphi)$. Lov\'asz's formula gives \[ \varphi(X)=\sup_{n\in\w}(m_n-2k_n+2)=\infty, \] so $X$ is positive. Let $A\subseteq X$ be positive. The nonempty traces $A\cap X_n$ partition $A$ into finite blocks, and for every $B\subseteq A$, \[ \varphi(B)=\sup_{n\in\w}\varphi\bigl(B\cap(A\cap X_n)\bigr). \] Moreover, $\sup_{n\in\w}\varphi(A\cap X_n)=\varphi(A)=\infty$. For each nonempty trace, the sequence $(S_i^n\cap A)_{i\frac{1}{3}, \] Theorem~\ref{thm:kneser-restr} shows that $X$ is positive and $\R\res X$ is hereditarily pathological. Fact~\ref{fact:restr} then gives pathology of $\R$. \end{proof} For completeness, we recall the following standard observation, obtained by embedding a disjoint union of finite complete graphs into $R$ and using the nonpathology of $\EDfin$. \begin{fact}\label{fact:edfin} There is an $\R$-positive set $Y$ with \[ \R\res Y\cong\EDfin =\left\{A\subseteq\bigsqcup_{n\geq1}V_n\mid \sup_{n\geq1}|A\cap V_n|<\infty\right\},\qquad |V_n|=n. \] This restriction is nonpathological, so $\R$ is not hereditarily pathological. \end{fact} \begin{proof} The homogeneous-set ideal of $\bigsqcup_{n\geq1}K_n$ is $\EDfin$, and it is nonpathological, by \cite[Examples~3.5 and~4.5]{MMU}; see also \cite[Example~5.26]{LOU}. Universality realizes this disjoint sum as $R[Y]$. Equation~\eqref{eq:induced} identifies its ideal with $\R\res Y$. The block sizes are unbounded, so $Y\notin\EDfin$ under the identification and $Y\in\R^+$. Fact~\ref{fact:restr} also shows that every positive restriction inside $Y$ is nonpathological. \end{proof} Corollary~\ref{cor:kneser-witness} and Fact~\ref{fact:edfin} together give the two local behaviors of $\R$. \begin{corollary}\label{cor:portrait} The random graph ideal $\R$ is pathological and has positive sets $X,Y$ such that $\R\res X$ is hereditarily pathological and $\R\res Y\cong\EDfin$ is nonpathological. \end{corollary} The Kneser construction also connects the pathological local behavior with Solecki's ideal, which appears in Hru\v{s}\'ak's measure dichotomy for analytic $P$-ideals \cite[Corollary~5.6]{Hrusak}. We now give an explicit comparison in the Kat\v{e}tov order. Let $m$ be the usual product probability measure on $2^\w$, and put \[ \Omega=\left\{U\in\mathrm{Clop}(2^\w)\mid m(U)=\frac{1}{2}\right\}. \] Solecki's ideal $\mathcal S$ on $\Omega$ is generated by the sets $I_x=\{U\in\Omega\mid x\in U\}$, for $x\in2^\w$. Recall that, for ideals $\mathcal I$ on $U$ and $\mathcal J$ on $V$, the relation $\mathcal I\leq_K\mathcal J$ means that there is a map $f:V\to U$ such that $f^{-1}(A)\in\mathcal J$ for every $A\in\mathcal I$. Let $\mathcal S_{1/4}$ denote the analogous ideal on the clopen subsets of $2^\w$ of measure $1/4$. The argument in the proof of \cite[Theorem~3.1]{HM} gives $\mathcal S\leq_K\mathcal S_{1/4}$. \begin{proposition}\label{prop:solecki} There is an $\R$-positive set $X$ such that $\R\res X$ is hereditarily pathological and \[ \mathcal S\leq_K\R\res X. \] \end{proposition} \begin{proof} We will code the vertices of the Kneser blocks by clopen sets of measure $1/4$ to witness $\mathcal S_{1/4}\leq_K\R\res X$. By universality, choose an induced copy of \[ G=\bigsqcup_{n\in\w}\KG(2^{n+2},2^n) \] on a set $X$. Since $2^n/2^{n+2}=1/4$ and $2^{n+2}-2\cdot2^n=2^{n+1}\to\infty$, Theorem~\ref{thm:kneser-restr} shows that $X\in\R^+$ and $\R\res X$ is hereditarily pathological. Write $X=\bigsqcup_{n\in\w}X_n$, where $X_n$ is the vertex set of the $n$th block. Label the points of $X_n$ by pairs $(n,a)$, with $a\in[2^{n+2}]^{2^n}$, so that two distinct points $(n,a)$ and $(n,b)$ are adjacent exactly when $a\cap b=\emptyset$. For a finite binary sequence $s$, write $[s]=\{x\in2^\w\mid s\subseteq x\}$. Enumerate the binary sequences of length $n+2$ as $(s_i^n)_{i<2^{n+2}}$. For $v=(n,a)\in X$, put \[ f(v)=\bigcup_{i\in a}[s_i^n]. \] The cylinders in this union are pairwise disjoint and each has measure $2^{-(n+2)}$, so $m(f(v))=1/4$. Fix $x\in2^\w$. For each $n\in\w$, let $i_x(n)$ be the unique index such that $s_{i_x(n)}^n\subseteq x$. The inverse image under $f$ of the generator $\{U\in\mathrm{Clop}(2^\w)\mid m(U)=1/4,\ x\in U\}$ is \[ \bigcup_{n\in\w}\{(n,a)\in X_n\mid i_x(n)\in a\}. \] This set is independent: its trace on each block is a star, and there are no edges between distinct blocks. Thus, $f$ witnesses $\mathcal S_{1/4}\leq_K\R\res X$. Together with $\mathcal S\leq_K\mathcal S_{1/4}$, this proves the assertion. \end{proof} Together with Corollary~\ref{cor:portrait}, this comparison relates the contrasting local behavior of $\R$ to two familiar ideals: $\EDfin$ on the nonpathological side and Solecki's ideal on the pathological side. The former occurs as a positive restriction of $\R$, up to isomorphism, while the latter lies below a hereditarily pathological positive restriction in the Kat\v{e}tov order. \section{Compact generators and least nonpathological extensions}\label{sec:generators} For an lscsm $\varphi$ on $V$, its \emph{nonpathological envelope} is\footnote{Recall that $\mathcal M(V,\varphi)$ is the family of measures $\mu$ on $V$ such that $\mu(B)\leq\varphi(B)$ for every $B\subseteq V$; see Section~\ref{sec:prelim}.} \[ \widehat\varphi(A)=\sup_{\mu\in\mathcal M(V,\varphi)}\mu(A). \] Since $\widehat\varphi\leq\varphi$, we have $\FIN(\varphi)\subseteq\FIN(\widehat\varphi)$. Moreover, $\FIN(\widehat\varphi)$ is proper exactly when $\widehat\varphi$ is unbounded. \begin{proposition}[Envelope facts; {\cite[\S3.1]{MMU}}]\label{prop:envelope} Let $\varphi$ be an lscsm on $V$. \begin{enumerate} \item[(i)] The envelope $\widehat\varphi$ is the largest nonpathological lscsm dominated by $\varphi$. In particular, $\varphi$ is nonpathological exactly when $\widehat\varphi=\varphi$. \item[(ii)] The supremum defining $\widehat\varphi$ is unchanged if only finitely supported measures are used. \item[(iii)] For every finite $F\subseteq V$, \begin{equation}\label{eq:finite-envelope} \widehat\varphi(F)=\max\left\{\nu(F)\mathrel{}\middle|\mathrel{} \begin{array}{l} \nu\text{ is a measure on }F,\\ \nu(B)\leq\varphi(B)\text{ for every }B\subseteq F \end{array} \right\}. \end{equation} Every measure $\nu$ on $F$ satisfying $\nu(B)\leq\varphi(B)$ for every $B\subseteq F$ extends by zero to a measure on $V$ dominated by $\varphi$. \end{enumerate} \end{proposition} \begin{proof} Assertion (i) follows from \cite[\S3.1]{MMU}, originating in \cite[p.~21]{Farah}. For (ii), let $\mu$ be a measure dominated by $\varphi$ and, for each finite $F\subseteq V$, define $\mu_F(B)=\mu(B\cap F)$. Then $\mu_F$ has finite support and \[ \mu_F(B)=\mu(B\cap F)\leq\varphi(B\cap F)\leq\varphi(B), \] so $\mu_F\leq\varphi$. Moreover, for every $A\subseteq V$, \[ \mu(A)=\sup_{F\in[A]^{<\omega}}\mu(F)=\sup_{F\in[A]^{<\omega}}\mu_F(A). \] Hence, restricting the supremum defining $\widehat\varphi$ to finitely supported measures does not change its value. For (iii), every measure dominated by $\varphi$ restricts to a measure $\nu$ on $F$ satisfying $\nu(B)\leq\varphi(B)$ for every $B\subseteq F$. Conversely, if $\nu$ satisfies these conditions, its extension by zero to $V$ satisfies \[ \widetilde\nu(A)=\nu(A\cap F) \leq\varphi(A\cap F)\leq\varphi(A) \qquad(A\subseteq V). \] Thus, $\widetilde\nu\leq\varphi$. A measure $\nu$ on $F$ is determined by the values $\nu(\{x\})$, for $x\in F$. The conditions $\nu(B)\leq\varphi(B)$ define a closed subset of the compact finite product $\prod_{x\in F}[0,\varphi(\{x\})]$. Since $\nu(F)=\sum_{x\in F}\nu(\{x\})$ is continuous in these coordinates, the maximum is attained. \end{proof} We now use compact hereditary families to represent ideals by covering submeasures. The following lemma is folklore, but we could not find a reference for its proof, so we include one here. \begin{lemma}\label{lem:cov} If $\mathcal C\subseteq2^V$ is closed and hereditary and contains all singletons, then the function \[ \varphi_{\mathcal C}(A) = \min\left\{ r\in\w\mathrel{}\middle|\mathrel{} A\subseteq\bigcup_{j\in r}K_j \text{ for some }(K_j)_{j\in r}\in\mathcal C^r \right\}, \qquad \min\emptyset=\infty, \] is an lscsm and $\FIN(\varphi_{\mathcal C})=\langle\mathcal C\rangle$. \end{lemma} \begin{proof} Monotonicity and the value at the empty set are immediate. If $A$ and $B$ can be covered by $r$ and $s$ members of $\mathcal C$, respectively, then $A\cup B$ can be covered by $r+s$ members, so $\varphi_{\mathcal C}$ is subadditive. Since $\mathcal C$ contains all singletons, $\varphi_{\mathcal C}$ is finite on finite sets. Moreover, \[ \FIN(\varphi_{\mathcal C})=\langle\mathcal C\rangle \] directly from the definition. It remains to prove lower semicontinuity. For $r\in\w$, let $E_r=\{A\subseteq V\mid\varphi_{\mathcal C}(A)\leq r\}$. We will show that each $E_r$ is closed. Since $\mathcal C$ is closed in the compact space $2^V$, the product $2^V\times\mathcal C^r$ is compact. Moreover, \[ \left\{(A,(K_j)_{j\in r})\in2^V\times\mathcal C^r\mid A\subseteq\bigcup_{j\in r}K_j\right\} \] is closed: failure of the covering condition is witnessed by some $x\in V$ with $x\in A$ and $x\notin K_j$ for every $j\in r$. Hence, this set is compact, and so its projection $E_r$ is compact, therefore closed. It is also hereditary. Consequently, \[ A\in E_r \quad\Longleftrightarrow\quad F\in E_r\text{ for every finite }F\subseteq A. \] Thus, \[ \varphi_{\mathcal C}(A) = \sup\{\varphi_{\mathcal C}(F)\mid F\subseteq A\text{ finite}\}, \] which is lower semicontinuity. \end{proof} \begin{definition}\label{def:fractional-cover} Let $F$ be a finite set and $\mathcal A\subseteq\mathcal P(F)$. A fractional cover of $F$ by members of $\mathcal A$ is a function $w:\mathcal A\to[0,\infty)$ satisfying \[ \sum_{\substack{K\in\mathcal A\\v\in K}}w(K)\geq1 \qquad(v\in F). \] We call $w(K)$ the weight of $K$ and $\sum_{K\in\mathcal A}w(K)$ the total weight of the cover. \end{definition} For the rest of this section, fix a compact hereditary family $\mathcal C\subseteq2^V$ containing all singletons. For $F\subseteq V$, put $\mathcal C_F=\mathcal C\cap\mathcal P(F)=\{K\cap F\mid K\in\mathcal C\}$, where the last equality holds as $\mathcal C$ is hereditary. Our aim is to show that $\FIN(\widehat{\varphi_{\mathcal C}})$ is contained in every nonpathological $F_\sigma$ ideal containing $\langle\mathcal C\rangle$ (Theorem~\ref{thm:least-extension}). As a first step, the following proposition computes $\widehat{\varphi_{\mathcal C}}(F)$ for finite $F$ as the minimum total weight of a fractional cover by members of $\mathcal C_F$. \begin{proposition}[Finite covering duality]\label{prop:cover-duality} For every $F\in[V]^{<\w}\setminus\{\emptyset\}$, \begin{equation}\label{eq:cover-duality} \widehat{\varphi_{\mathcal C}}(F) =\min\left\{\sum_{K\in\mathcal C_F}w(K)\mathrel{}\middle|\mathrel{} w:\mathcal C_F\to[0,\infty),\quad \sum_{\substack{K\in\mathcal C_F\\v\in K}}w(K)\geq1\ (v\in F)\right\}. \end{equation} Moreover, there is a measure $\nu_F$ on $F$ such that $\nu_F(F)$ equals this minimum and $\nu_F(K)\leq1$ for every $K\in\mathcal C_F$. Its extension $\mu_F(B)=\nu_F(B\cap F)$ is dominated by $\varphi_{\mathcal C}$ on all of $V$. \end{proposition} \begin{proof} Assigning weight $1$ to each singleton and $0$ to the other members of $\mathcal C_F$ gives a fractional cover of $F$, so the minimum in \eqref{eq:cover-duality} is finite. By finite covering duality \cite[Theorem~2.1]{CJ}, this minimum equals the maximum of $\nu(F)$ over measures on $F$ such that $\nu(K)\leq1$ for every $K\in\mathcal C_F$. We claim that a measure $\nu$ on $F$ satisfies $\nu(K)\leq1$ for every $K\in\mathcal C_F$ if and only if $\nu(B)\leq\varphi_{\mathcal C}(B)$ for every $B\subseteq F$. Indeed, if $\nu\leq\varphi_{\mathcal C}$ on $\mathcal P(F)$, then $\nu(K)\leq\varphi_{\mathcal C}(K)\leq1$ for $K\in\mathcal C_F$. Conversely, suppose that $\nu(K)\leq1$ for every $K\in\mathcal C_F$. For $B\subseteq F$, put $r=\varphi_{\mathcal C}(B)$, which is finite since $B$ is finite. Since $\mathcal C$ is hereditary, we can choose a cover of $B$ by $r$ members $K_j$ of $\mathcal C_F$. Then \[ \nu(B)\leq\sum_{j0$ such that $\lambda(E_v)\geq\delta$ for every $v\in A$. For $m\in\w$, put \[ D_m=\{K\in\mathcal C_A\mid\psi(K)\leq m\}. \] The sequence $(D_m\mid m\in\w)$ is increasing and covers $\mathcal C_A$, since $\psi$ is finite on every member of $\mathcal C_A$. Each $D_m$ is closed by lower semicontinuity. By continuity from below, $\lambda(D_m)\to1$, so $\lambda(\mathcal C_A\setminus D_m)=1-\lambda(D_m)\to0$. Choose $m$ such that $\lambda(\mathcal C_A\setminus D_m)<\delta/2$. For every $v\in A$, we have \begin{equation}\label{eq:generator-coverage} \lambda(D_m\cap E_v) =\lambda(E_v)-\lambda(E_v\setminus D_m) \geq\delta-\lambda(\mathcal C_A\setminus D_m) \geq\frac{\delta}{2}. \end{equation} Let $\mu$ be a measure on $V$ dominated by $\psi$, with finite support $F$. We will show that $\mu(A)\leq2m/\delta$. Note that $\mu(K)=\sum_{v\in F\cap A}\mu(\{v\})\mathbf1_{E_v}(K)$ for $K\in\mathcal C_A$. Also, $\mu(K)\leq\psi(K)\leq m$ for $K\in D_m$. Multiplying \eqref{eq:generator-coverage} by $\mu(\{v\})$ and summing over $v\in F\cap A$, we obtain \[ \frac{\delta}{2}\mu(A) \leq\sum_{v\in F\cap A}\mu(\{v\})\lambda(D_m\cap E_v) =\int_{D_m}\mu(K)\,d\lambda(K) \leq m\lambda(D_m)\leq m. \] The equality follows by integrating the finite sum above. Thus, $\mu(A)\leq2m/\delta$ for every finitely supported measure $\mu\leq\psi$. Taking the supremum and using Proposition~\ref{prop:envelope}(ii), we obtain $\widehat\psi(A)\leq2m/\delta$. Since $\psi$ is nonpathological, $\psi(A)=\widehat\psi(A)<\infty$. \end{proof} Every $F_\sigma$ ideal is generated by a compact hereditary family containing all singletons; this fact is also recorded in \cite[Remark~8.6(3)]{Beros}. To see this directly, identify its underlying set with $\w$ and write $\mathcal I=\FIN(\varphi)$ for an lscsm $\varphi$, by Mazur's theorem \cite{Mazur}. Put \[ \mathcal C_n=\{B\subseteq\w\setminus n\mid\varphi(B)\leq n\}, \qquad \mathcal C=[\w]^{\leq1}\cup\bigcup_{n\in\w}\mathcal C_n. \] Each $\mathcal C_n$ is compact, and every neighborhood of $\emptyset$ contains $\mathcal C_n$ for all sufficiently large $n$. Since $[\w]^{\leq1}$ is also compact, $\mathcal C$ is compact. It is hereditary, contains all singletons and is contained in $\mathcal I$. If $B\in\mathcal I$, choose $n\in\w$ with $n\geq\varphi(B)$. Then $B\setminus n\in\mathcal C_n$, while $B\cap n$ is finite, so $B\in\langle\mathcal C\rangle$. Thus, $\mathcal I=\langle\mathcal C\rangle$, and the following theorem applies to every $F_\sigma$ ideal. \begin{theorem}[Least nonpathological extension]\label{thm:least-extension} Suppose $\mathcal I=\langle\mathcal C\rangle$ is proper, and put \[ \mathcal J_{\mathcal I}=\FIN(\widehat{\varphi_{\mathcal C}}). \] Then: \begin{enumerate} \item[(i)] If $\mathcal J_{\mathcal I}$ is proper, it is the least nonpathological $F_\sigma$ ideal containing $\mathcal I$. \item[(ii)] If $\mathcal J_{\mathcal I}=\mathcal P(V)$, no proper nonpathological $F_\sigma$ ideal contains $\mathcal I$. \end{enumerate} Moreover, $\mathcal J_{\mathcal I}$ does not depend on the choice of $\mathcal C$. \end{theorem} \begin{proof} Let $\mathcal L=\FIN(\psi)$ be a proper nonpathological $F_\sigma$ ideal containing $\mathcal I$, with $\psi$ nonpathological. Each $K\in\mathcal C$ belongs to $\mathcal I$, so $\psi(K)<\infty$. Lemma~\ref{lem:cover-comparison} gives $\mathcal J_{\mathcal I}\subseteq\mathcal L$. By Proposition~\ref{prop:envelope}(i), $\widehat{\varphi_{\mathcal C}}$ is a nonpathological lscsm. Since $\widehat{\varphi_{\mathcal C}}\leq\varphi_{\mathcal C}$, we have $\mathcal I\subseteq\mathcal J_{\mathcal I}$. If $\mathcal J_{\mathcal I}$ is proper, it is therefore a nonpathological $F_\sigma$ extension of $\mathcal I$, and the preceding paragraph shows that it is the least one. If $\mathcal J_{\mathcal I}=\mathcal P(V)$, the inclusion $\mathcal J_{\mathcal I}\subseteq\mathcal L$ forces $\mathcal L=\mathcal P(V)$, so no such proper extension exists. Thus, $\mathcal J_{\mathcal I}$ is the least such extension when one exists, and is $\mathcal P(V)$ otherwise. This description depends only on $\mathcal I$, not on the choice of $\mathcal C$. \end{proof} For any lscsm $\varphi$ representing $\mathcal I$, its envelope is nonpathological and $\mathcal I\subseteq\FIN(\widehat\varphi)$. This alone does not show that $\FIN(\widehat\varphi)$ is contained in every proper nonpathological $F_\sigma$ extension of $\mathcal I$. For the covering representation, Lemma~\ref{lem:cover-comparison} gives precisely this inclusion. The second alternative is precisely super-pathology of $\mathcal I$, which we study in Section~\ref{sec:super}. For $g:V\to(0,\infty)$ with $\sum_{v\in V}g(v)=\infty$, write $\mathcal I_g=\{A\subseteq V\mid\sum_{v\in A}g(v)<\infty\}$ for the associated proper summable ideal. Filip\'ow and Tryba \cite[Theorem~11.2]{FT} and, independently, Figueroa-Sierra and Hru\v{s}\'ak \cite[Corollary~6.1.2]{FigueroaThesis} proved that every nonpathological $F_\sigma$ ideal is an intersection of summable ideals. Using the least-extension theorem, we identify the intersection of all proper summable ideals containing an arbitrary $F_\sigma$ ideal and obtain the converse. \begin{corollary}\label{cor:summable-intersection} For every proper $F_\sigma$ ideal $\mathcal I$ on $V$, \[ \mathcal J_{\mathcal I} =\bigcap\{\mathcal I_g\mid\mathcal I\subseteq\mathcal I_g, \ \mathcal I_g\text{ is a proper summable ideal on }V\}, \] where the intersection of the empty family is $\mathcal P(V)$. In particular, an $F_\sigma$ ideal is nonpathological if and only if it is an intersection of summable ideals. \end{corollary} \begin{proof} Let $\mathcal L$ denote the intersection on the right-hand side. Every proper summable ideal is nonpathological and $F_\sigma$, since it is represented by the measure $\mu_g(A)=\sum_{v\in A}g(v)$. Thus, Theorem~\ref{thm:least-extension} gives $\mathcal J_{\mathcal I}\subseteq\mathcal L$. We prove the reverse inclusion. If $\mathcal J_{\mathcal I}=\mathcal P(V)$, it is immediate. Otherwise, $\mathcal J_{\mathcal I}$ is a proper nonpathological $F_\sigma$ ideal, so the theorem of Filip\'ow--Tryba and Figueroa-Sierra--Hru\v{s}\'ak represents it as an intersection of proper summable ideals. Each ideal in this representation contains $\mathcal I$ and therefore occurs among those defining $\mathcal L$. Thus, $\mathcal L$ is contained in each of them, and hence in their intersection $\mathcal J_{\mathcal I}$. Finally, if $\mathcal I$ is an intersection of summable ideals, it equals the intersection of all proper summable ideals containing it. Hence, $\mathcal I=\mathcal L=\mathcal J_{\mathcal I}$ is nonpathological. \end{proof} Corollary~\ref{cor:summable-intersection} and the characterization of Filip\'ow and Tryba \cite[Proposition~11.4]{FT} show that an $F_\sigma$ ideal $\mathcal I$ is nonpathological if and only if every positive restriction $\mathcal I\res A$ is Kat\v{e}tov below some summable ideal. We next consider matrix extensions. A nonnegative matrix $B=(b_{n,v}\mid n\in\w,\,v\in V)$ with finite row sums is \emph{regular} if \[ \lim_{n\to\infty}\sum_{v\in V}b_{n,v}=1, \qquad \lim_{n\to\infty}b_{n,v}=0 \quad(v\in V). \] Thus, the row sums converge to $1$, while each column converges to $0$. In particular, the row sums are uniformly bounded. For $A\subseteq V$, put $\mu_n(A)=\sum_{v\in A}b_{n,v}$. The associated \emph{matrix ideal} is \[ \mathcal M_B=\left\{A\subseteq V\mid\lim_{n\to\infty}\mu_n(A)=0\right\}; \] see \cite[\S8]{FT}. Convergence to zero in each column gives $\mu_n(F)\to0$ for every finite $F\subseteq V$, so $\mathcal M_B$ contains all finite sets. Since $\mu_n(V)\to1$, we have $V\notin\mathcal M_B$, so the ideal is proper. \begin{proposition}\label{prop:matrix-extension} Let $\mathcal I$ be a proper $F_\sigma$ ideal on $V$. Every matrix ideal containing $\mathcal I$ also contains $\mathcal J_{\mathcal I}$. \end{proposition} \begin{proof} Suppose $\mathcal I\subseteq\mathcal M_B$. By the construction preceding Theorem~\ref{thm:least-extension}, choose a compact hereditary family $\mathcal C$ containing all singletons and generating $\mathcal I$. Fix $A\in\mathcal J_{\mathcal I}$. By Lemma~\ref{lem:probability-cover}, there are a probability $\lambda$ on $\mathcal C_A$ and $\delta>0$ such that $\lambda(E_v)\geq\delta$ for every $v\in A$. For each $K\in\mathcal C_A$, we have $K\in\mathcal I\subseteq\mathcal M_B$, so $\mu_n(K)\to0$. The functions \[ K\longmapsto\mu_n(K) =\sum_{v\in A}b_{n,v}\mathbf1_{E_v}(K) \] are Borel, since each $E_v$ is clopen and the partial sums are continuous. They are bounded by $L=\sup_{n\in\w}\mu_n(V)<\infty$. Since $\lambda$ is a probability, dominated convergence gives \[ \int_{\mathcal C_A}\mu_n(K)\,d\lambda(K)\longrightarrow0. \] Moreover, \[ \delta\,\mu_n(A) \leq\sum_{v\in A}b_{n,v}\lambda(E_v) =\int_{\mathcal C_A}\mu_n(K)\,d\lambda(K). \] The inequality uses $\lambda(E_v)\geq\delta$, and the equality follows from Tonelli's theorem, since all summands are nonnegative. Thus, $\mu_n(A)\to0$, so $A\in\mathcal M_B$. This proves $\mathcal J_{\mathcal I}\subseteq\mathcal M_B$. \end{proof} Every nonpathological $F_\sigma$ ideal is an intersection of matrix ideals \cite[Theorem~8.5]{FT}. Together with Proposition~\ref{prop:matrix-extension}, this identifies the intersection of all matrix ideals containing a given $F_\sigma$ ideal. \begin{corollary}\label{cor:matrix-intersection} For every proper $F_\sigma$ ideal $\mathcal I$ on $V$, \[ \mathcal J_{\mathcal I} =\bigcap\{\mathcal M\mid\mathcal I\subseteq\mathcal M, \ \mathcal M\text{ is a matrix ideal on }V\}, \] where the intersection of the empty family is $\mathcal P(V)$. In particular, the following are equivalent: \begin{enumerate} \item[(i)] $\mathcal I$ is nonpathological. \item[(ii)] $\mathcal I$ is an intersection of summable ideals. \item[(iii)] $\mathcal I$ is an intersection of matrix ideals. \end{enumerate} \end{corollary} \begin{proof} Let $\mathcal L$ denote the intersection on the right-hand side. Proposition~\ref{prop:matrix-extension} gives $\mathcal J_{\mathcal I}\subseteq\mathcal L$. We prove the reverse inclusion. If $\mathcal J_{\mathcal I}=\mathcal P(V)$, it is immediate. Otherwise, $\mathcal J_{\mathcal I}$ is a proper nonpathological $F_\sigma$ ideal, so it is an intersection of matrix ideals. Each ideal in this representation contains $\mathcal I$ and therefore occurs among those defining $\mathcal L$. Thus, $\mathcal L$ is contained in each of them, and hence in their intersection $\mathcal J_{\mathcal I}$. The equivalence of (i) and (ii) is Corollary~\ref{cor:summable-intersection}, and the representation recalled above gives (i)$\Rightarrow$(iii). For (iii)$\Rightarrow$(i), suppose that $\mathcal I$ is an intersection of matrix ideals. It then equals the intersection of all matrix ideals containing it, so the identity just proved gives $\mathcal I=\mathcal J_{\mathcal I}$. Thus, $\mathcal I$ is nonpathological. \end{proof} The equivalence of (i)--(iii) answers both converses asked in \cite[Question~11.6]{FT}. The inclusion in Proposition~\ref{prop:matrix-extension} also applies to nonpathological analytic $P$-ideal extensions. Recall that an ideal $\mathcal L$ is a $P$-ideal if, for every sequence $(A_n)_{n\in\w}$ in $\mathcal L$, there is $A\in\mathcal L$ such that $A_n\setminus A$ is finite for every $n$. Solecki's representation theorem \cite{Solecki} provides the counterpart of Mazur's theorem for analytic $P$-ideals. On $\w$, these are precisely the proper ideals of the form \[ \Exh(\psi) =\left\{A\subseteq\w\mid\lim_{n\to\infty}\psi(A\setminus n)=0\right\}, \] where $\psi$ is an lscsm. Such an ideal is called nonpathological if it admits this representation with $\psi$ nonpathological \cite[Definition~8.3]{FT}. For analytic $P$-ideals, Tryba \cite[Corollary~3.12]{Tryba2024} proved that nonpathology is equivalent to being an intersection of matrix ideals, with nonpathology understood through $\Exh$. For an $F_\sigma$ $P$-ideal, the two definitions of nonpathology use different representations. The last assertion below shows that a nonpathological $\Exh$ representation guarantees the existence of a nonpathological $\FIN$ representation. \begin{corollary}\label{cor:analytic-p-extension} Let $\mathcal I$ be a proper $F_\sigma$ ideal on $\w$. Every nonpathological analytic $P$-ideal containing $\mathcal I$ also contains $\mathcal J_{\mathcal I}$. In particular, if an $F_\sigma$ ideal is nonpathological as an analytic $P$-ideal, then it is nonpathological as an $F_\sigma$ ideal. \end{corollary} \begin{proof} Let $\mathcal L$ be a nonpathological analytic $P$-ideal containing $\mathcal I$. By \cite[Theorem~8.5]{FT}, $\mathcal L$ is an intersection of matrix ideals. Each ideal in this representation contains $\mathcal I$, so Proposition~\ref{prop:matrix-extension} shows that it contains $\mathcal J_{\mathcal I}$. Taking their intersection gives $\mathcal J_{\mathcal I}\subseteq\mathcal L$. For the last assertion, take $\mathcal L=\mathcal I$. Then $\mathcal I\subseteq\mathcal J_{\mathcal I}\subseteq\mathcal I$, so $\mathcal I=\mathcal J_{\mathcal I}$ is nonpathological as an $F_\sigma$ ideal. \end{proof} By the representation results recalled in \cite[\S2, after Definition~2.2]{LOU}, nonpathological analytic $P$-ideals are precisely the ideals representable by unconditional convergence in a Banach space, while nonpathological $F_\sigma$ ideals are precisely those representable by bounded finite subsums. Thus, the last assertion answers \cite[Question~2.4]{LOU} affirmatively. \section{Fractional coloring and graphs with no infinite clique}\label{sec:coloring} We apply the results of Section~\ref{sec:generators} to the family of independent sets of a graph. The envelope identity holds for every countable graph. When the graph has no infinite clique, independent sets generate its homogeneous-set ideal, so the least-extension theorem applies to that ideal. We also use this hypothesis to extract pairwise disjoint finite subgraphs with no edges between them and obtain a criterion for nonpathology. For a graph $G$, write $\mathrm{Ind}(G)$ for the family of its independent sets. \begin{definition}\label{def:frac} For a finite graph $K$, a fractional coloring of $K$ is a fractional cover of $V(K)$ by members of $\mathrm{Ind}(K)$. Thus, its weights satisfy \[ \sum_{\substack{I\in\mathrm{Ind}(K)\\v\in I}}w(I)\geq1 \qquad(v\in V(K)). \] The fractional chromatic number $\chif(K)$ is the infimum of the total weights of these covers. Put $\chif(\emptyset)=0$. \end{definition} \begin{fact}[Fractional colorings and covering sequences]\label{fact:frac} \widowpenalty=10000 For every nonempty finite graph $K$, there is an optimal fractional coloring with rational weights. Consequently, some nonempty finite sequence of independent sets covers each vertex in at least a $1/\chif(K)$ fraction of its terms. \end{fact} \begin{proof} The existence of an optimal rational fractional coloring is a standard consequence of finite linear programming \cite[\S1.3]{SU}. Choose such a coloring $w$ and a positive integer $N$ such that $Nw(I)$ is an integer for every $I\in\mathrm{Ind}(K)$. Form a sequence $(I_j)_{j0. \] For every $v\in V(K)$, the fractional coloring constraint gives \[ |\{jk$ for every $k$ and there are no edges between distinct $B_k$. \end{lemma} \begin{proof} In this proof, call $U\subseteq V$ unbounded (with respect to $\varphi$ and $\mathcal A$) if \[ \sup\{\varphi(F)\mid F\in\mathcal A,\ F\subseteq U\}=\infty. \] For $v\in U$, write $N_U(v)$ for its neighborhood in $H[U]$. There is an unbounded set $U$ such that $N_U(v)$ is not unbounded for any $v\in U$. Indeed, otherwise we could start with $U_0=V$ and choose recursively $v_j\in U_j$ such that $U_{j+1}=N_{U_j}(v_j)$ is unbounded. The vertices $v_j$ would be distinct and pairwise adjacent, contrary to the hypothesis on $H$. Fix such a set $U$. For each $v\in U$, choose $C_v<\infty$ such that \[ \varphi(F)\le C_v \qquad(F\in\mathcal A,\ F\subseteq N_U(v)). \] Suppose that $B_i$ has been chosen for $ik+\varphi(P)+\sum_{v\in P}C_v. \] Set \[ B_k=F\setminus\left(P\cup\bigcup_{v\in P}N_U(v)\right). \] Then $B_k\cap P=\emptyset$ and there are no edges between $B_k$ and $P$; also, as $\mathcal{A}$ is hereditary, $B_k\in\mathcal A$. Each $F\cap N_U(v)$ also belongs to $\mathcal A$, so monotonicity and subadditivity give \[ \varphi(F) \le \varphi(B_k)+\varphi(P)+\sum_{v\in P}\varphi(F\cap N_U(v)) \le \varphi(B_k)+\varphi(P)+\sum_{v\in P}C_v. \] Consequently, $\varphi(B_k)>k$. In particular, $B_k$ is nonempty. \end{proof} For a countable graph $H$ with no infinite clique, every clique is finite, so a finite homogeneous cover can be replaced by a finite independent cover. Consequently, \[ \mathcal I_H=\langle\hg(H)\rangle=\FIN(\varphi_\chi^H). \] We characterize nonpathology by asking whether bounded fractional chromatic number forces bounded chromatic number uniformly over the finite induced subgraphs of $H$. \begin{theorem}\label{thm:nonpath-char} Let $H=(V,E)$ be a countable graph with no infinite clique and with $\chi(H)=\infty$, and put $\mathcal I_H=\langle\hg(H)\rangle$. Write $\chromsub$ and $\fracsub$ for its chromatic and fractional submeasures. The following are equivalent: \begin{enumerate} \item[(i)] $\mathcal I_H$ is nonpathological; \item[(ii)] $\FIN(\chromsub)=\FIN(\fracsub)$; \item[(iii)] for every $C>0$ there is $M_C<\infty$ such that, for every finite $F\subseteq V$, \begin{equation}\label{eq:hereditary-bound} \chif(H[F])\le C \quad\Longrightarrow\quad \chi(H[F])\le M_C. \end{equation} \end{enumerate} If these conditions fail, there is $A\in\mathcal I_H^+$ such that $H[A]$ is a disjoint union of nonempty finite graphs with unbounded chromatic numbers and uniformly bounded fractional chromatic numbers. Moreover, $\mathcal I_H\res A$ is hereditarily pathological. \end{theorem} \begin{proof} (i)$\Leftrightarrow$(ii). Suppose that $\mathcal I_H$ is nonpathological. Apply Theorem~\ref{thm:least-extension} to the family of independent sets, using $\mathcal I_H$ itself as the nonpathological extension. It gives $\FIN(\fracsub)\subseteq\mathcal I_H=\FIN(\chromsub)$, and the reverse inclusion always holds. Conversely, if (ii) holds, Theorem~\ref{thm:env} shows that $\fracsub$ is a nonpathological lscsm representing $\mathcal I_H$. (iii)$\Rightarrow$(ii). The inclusion $\FIN(\chromsub)\subseteq\FIN(\fracsub)$ follows from $\fracsub\le\chromsub$. For a nonempty $A\subseteq V$ with $\fracsub(A)<\infty$, put $C=\fracsub(A)>0$. Every finite $F\subseteq A$ then satisfies $\chif(H[F])\le C$, and hence $\chi(H[F])\le M_C$. Lower semicontinuity of $\chromsub$ gives $\chromsub(A)\le M_C<\infty$. To prove (ii)$\Rightarrow$(iii) by contraposition and establish the final assertion, suppose that (iii) fails for some $C>0$. Put \[ \mathcal A=\{F\in[V]^{<\w}\mid\chif(H[F])\le C\}, \qquad \varphi(F)=\chi(H[F]). \] The family $\mathcal A$ is hereditary, as fractional chromatic number is monotone under induced restriction. The function $\varphi:[V]^{<\w}\to[0,\infty)$ is monotone and subadditive, with $\varphi(\emptyset)=0$. Since (iii) fails for $C$, for every $M<\infty$ there is $F\in\mathcal A$ with $\varphi(F)>M$. Lemma~\ref{lem:anticomplete} therefore gives pairwise disjoint sets $B_k$, with no edges between distinct sets, such that \[ \chi(H[B_k])>k,\qquad \chif(H[B_k])\le C. \] Put $A=\bigcup_{k\in\w}B_k$. Since $\chi(H[A])\geq\chi(H[B_k])>k$ for every $k$, we have $\chromsub(A)=\infty$. On the other hand, every nonempty finite $F\subseteq A$ meets only finitely many $B_k$. There are no edges between distinct $B_k$, so \eqref{eq:frac-disjoint} gives \[ \chif(H[F]) =\max_{\substack{k\in\w\\F\cap B_k\ne\emptyset}}\chif(H[F\cap B_k]) \leq C. \] Taking the supremum over finite $F\subseteq A$, we obtain $\fracsub(A)\leq C$. Thus, $A\in\FIN(\fracsub)\setminus\FIN(\chromsub)$, so (ii) fails. Moreover, $A\in\mathcal I_H^+$. It remains to prove hereditary pathology. Let $D\subseteq A$ with $D\notin\mathcal I_H$. Restricting homogeneous covers gives $\mathcal I_H\res D=\mathcal I_{H[D]}$. The graph $H[D]$ has no infinite clique, and its chromatic and fractional submeasures are the restrictions of $\chromsub$ and $\fracsub$ to $\mathcal P(D)$. Since \[ \chromsub(D)=\infty \qquad\text{and}\qquad \fracsub(D)\leq\fracsub(A)\leq C, \] condition (ii) fails for $H[D]$. By the equivalence (i)$\Leftrightarrow$(ii), already proved, $\mathcal I_{H[D]}$ is pathological. Thus, $\mathcal I_H\res A$ is hereditarily pathological. \end{proof} Let $H=(V,E)$ be a countable graph with no infinite clique, and let $\psi$ be an lscsm on $V$ that is finite on every independent set of $H$. Then there is $L>0$ such that $\psi(I)\leq L$ for every independent set $I$. Indeed, if these values were unbounded, lower semicontinuity would give unbounded values on finite independent sets. Lemma~\ref{lem:anticomplete}, applied to this family and to $\psi$ on finite sets, would give independent sets $B_k$ with $\psi(B_k)>k$ and no edges between distinct sets. Their union is independent and, by monotonicity, has infinite $\psi$-value, a contradiction. A cover by independent sets and subadditivity give $\psi\leq L\varphi_\chi^H$. If $\psi$ is nonpathological, taking envelopes and using Theorem~\ref{thm:env} gives \[ \psi=\widehat\psi \leq L\widehat{\varphi_\chi^H} =L\varphi_{\chi_f}^H. \] Thus, every nonpathological lscsm that is finite on the independent sets of $H$ is dominated on all subsets of $V$ by a single constant multiple of $\varphi_{\chi_f}^H$. We now specialize these results to graph blocks. For the remainder of this section, let $G=\bigsqcup_{n\in\w}G_n$, where the blocks are finite and nonempty, put $V=V(G)$ and $V_n=V(G_n)$, and assume $\sup_{n\in\w}\chi(G_n)=\infty$. We use $\chromsub$ and $\fracsub$ for the submeasures associated with this graph. The graph $G$ has no infinite clique. By Lemma~\ref{lem:transfer} and equation~\eqref{eq:fractional-blocks}, \[ \chromsub(A)=\sup_{n\in\w}\chi(G_n[A\cap V_n]), \qquad \fracsub(A)=\sup_{n\in\w}\chif(G_n[A\cap V_n]). \] Thus, $\mathcal I_G=\langle\hg(G)\rangle=\FIN(\chromsub)$. We call these proper ideals \emph{graph block ideals}. By Theorem~\ref{thm:nonpath-char}, $\mathcal I_G$ is nonpathological if and only if for every $C>0$ there is $M_C<\infty$ such that \[ \chif(G_n[B])\le C \quad\Longrightarrow\quad \chi(G_n[B])\le M_C \qquad(n\in\w,\ B\subseteq V_n). \] Indeed, it suffices to check condition~\eqref{eq:hereditary-bound} on subsets of individual blocks: if a finite $F\subseteq V$ satisfies $\chif(G[F])\le C$, then each nonempty $F\cap V_n$ has fractional chromatic number at most $C$, and hence chromatic number at most $M_C$. Since there are no edges between distinct blocks, $\chi(G[F])\le M_C$. The star covers of the Kneser blocks in Corollary~\ref{cor:kneser-witness} may now be read as fractional colorings: assigning weight $1/k$ to each star gives \begin{equation}\label{eq:kneser-fractional-bound} \chif(\KG(3k-1,k))\leq\frac{3k-1}{k}<3. \end{equation} Thus, the concrete witness behind Corollary~\ref{cor:kneser-witness} is an instance of the general obstruction isolated in Theorem~\ref{thm:nonpath-char}: uniformly bounded fractional chromatic number together with unbounded chromatic number. Farah's degree of pathology is defined through ratios of a submeasure to its envelope \cite[p.~21]{Farah}. For lscsms it is enough to consider finite subsets, as in \cite[\S3.1]{MMU}; see also \cite[Proposition~7.2]{FT}. To distinguish this version from the normalized ratio on the whole space used in \cite[\S5]{Hrusak}, write \[ P_{\rm fin}(\varphi)= \sup\left\{ \frac{\varphi(F)}{\widehat\varphi(F)} \mathrel{}\middle|\mathrel{} F\text{ finite and }\widehat\varphi(F)>0 \right\}. \] For any countable graph $H$, Theorem~\ref{thm:env} gives \[ P_{\rm fin}(\varphi_\chi^H) =\sup\left\{ \frac{\chi(H[F])}{\chif(H[F])} \mathrel{}\middle|\mathrel{} F\in[V(H)]^{<\w}\setminus\{\emptyset\} \right\}. \] For graph blocks, it suffices to consider subsets of individual blocks. Indeed, let $F\subseteq V$ be finite and nonempty. Since there are no edges between distinct blocks, choose $n$ such that $\chi(G[F])=\chi(G_n[F\cap V_n])$. By monotonicity of fractional chromatic number, \[ \frac{\chi(G[F])}{\chif(G[F])} \leq \frac{\chi(G_n[F\cap V_n])}{\chif(G_n[F\cap V_n])}. \] Conversely, every nonempty subset of a block is a finite subset of $V$. Taking suprema therefore gives \begin{equation}\label{eq:degree} P_{\rm fin}(\chromsub) = \sup\left\{ \frac{\chi(G_n[B])}{\chif(G_n[B])}\mathrel{}\middle|\mathrel{}\emptyset\ne B\subseteq V_n \text{ for some }n\in\w\right\}. \end{equation} Finiteness of $P_{\rm fin}(\chromsub)$ means that a single constant bounds $\chi/\chif$ on all nonempty induced subgraphs of the blocks. The nonpathology criterion in Theorem~\ref{thm:nonpath-char} allows this bound to depend on an upper bound $C$ for $\chif$. Indeed, for each fixed $C\geq1$, bounding $\chi$ is equivalent to bounding $\chi/\chif$ on the subgraphs with $1\leq\chif\leq C$. In particular, $P_{\rm fin}(\chromsub)<\infty$ implies that $\mathcal I_G$ is nonpathological. On a nonempty whole block, Hru\v{s}\'ak's degree is $\chi(G_n)/\chif(G_n)$;\footnote{Hru\v{s}\'ak's formulation is entirely in terms of submeasures and Kelley covering numbers. The appearance of $\chif$ here comes from our identification $\widehat{\chromsub}=\fracsub$ in Theorem~\ref{thm:env}.} its quantitative relation to Kelley coverings is given by \cite[Lemma~5.4]{Hrusak}. \begin{remark}[Complexity in the block class]\label{rem:complexity} Question~5.21 of \cite{LOU} asks, in particular, for the descriptive complexity of the colorings whose coloring ideal is nonpathological. Theorem~\ref{thm:nonpath-char} gives a partial answer for the graph block ideals considered here. Code nonempty finite graphs by natural numbers and give the space of sequences of such codes the product topology. Restrict attention to sequences with unbounded chromatic numbers. Within this class, nonpathology is expressed by \[ (\forall C\in\omega\setminus\{0\})(\exists M\in\omega)(\forall n\in\omega)\; R(C,M,n), \] where $R(C,M,n)$ says that every $B\subseteq V_n$ satisfying $\chif(G_n[B])\leq C$ also satisfies $\chi(G_n[B])\leq M$. For fixed $C,M,n$, this condition depends only on the finite graph coded by the $n$th coordinate, so it defines a clopen set. These quantifiers therefore give a $\Pi^0_3$ definition of nonpathology relative to this class. \end{remark} \section{The canonical envelope and further consequences}\label{sec:global} We now describe the least nonpathological $F_\sigma$ extension of the random graph ideal $\R$ using fractional cochromatic numbers of finite induced subgraphs. Theorem~\ref{thm:least-extension} gives this extension from the homogeneous generators once we show that the corresponding envelope is unbounded. We prove this unboundedness and show that the Kneser set from Corollary~\ref{cor:kneser-witness} witnesses the strict inclusion of $\R$ in the extension. We first describe the corresponding envelope for an arbitrary countable graph $G=(V,E)$. The family $\hg(G)$ of homogeneous sets is closed, hereditary and contains all singletons. Write $\varphi_Z^G=\varphi_{\hg(G)}$, so \[ \FIN(\varphi_Z^G)=\langle\hg(G)\rangle=\mathcal I_G. \] \begin{definition}\label{def:cocoloring} For a finite graph $K$, the cochromatic number $Z(K)$ is the least number of homogeneous sets covering $V(K)$. A fractional cocoloring of $K$ is a fractional cover of $V(K)$ by members of $\hg(K)$ (Definition~\ref{def:fractional-cover}). The fractional cochromatic number $Z_{\mathrm f}(K)$ is the infimum of the total weights of these covers. Put $Z(\emptyset)=Z_{\mathrm f}(\emptyset)=0$. \end{definition} As with fractional chromatic number, we extend the finite parameter by taking the supremum over finite induced subgraphs. For a countable graph $G$ and $A\subseteq V(G)$, put \[ \varphi_{Z_f}^G(A)=\sup_{F\in[A]^{<\w}}Z_{\mathrm f}(G[F]). \] Applying Proposition~\ref{prop:cover-duality} to $\hg(G)$ gives the following envelope identity; see also \cite[\S2, p.~3]{GKM} for the finite cochromatic duality. \begin{proposition}[Cochromatic envelope]\label{prop:cochromatic-envelope} For every countable graph $G$ and every $A\subseteq V(G)$, \[ \varphi_Z^G(A)=\sup_{F\in[A]^{<\w}}Z(G[F]),\qquad \widehat{\varphi_Z^G}(A)=\varphi_{Z_f}^G(A). \] \end{proposition} \begin{proof} For finite $F\subseteq V(G)$, the members of $\hg(G)$ contained in $F$ are exactly the homogeneous sets of $G[F]$, so $\varphi_Z^G(F)=Z(G[F])$. Since $\hg(G)$ is closed, hereditary and contains all singletons, Lemma~\ref{lem:cov} makes $\varphi_Z^G$ lower semicontinuous. This proves the first equality for arbitrary $A$. Fractional covers of $F$ by these homogeneous sets are precisely fractional cocolorings of $G[F]$. Proposition~\ref{prop:cover-duality} therefore gives $\widehat{\varphi_Z^G}(F)=Z_{\mathrm f}(G[F])$ for nonempty finite $F$; the equality also holds for $F=\emptyset$, when both values are zero. By Proposition~\ref{prop:envelope}, the envelope is lower semicontinuous, so \[ \widehat{\varphi_Z^G}(A) =\sup_{F\in[A]^{<\w}}Z_{\mathrm f}(G[F]) =\varphi_{Z_f}^G(A).\qedhere \] \end{proof} We next compare the cochromatic envelope with the chromatic envelope from Theorem~\ref{thm:env}. \begin{proposition}\label{prop:chromatic-cochromatic} Suppose $\mathcal I_G$ is proper and $H$ is either $G$ or its complement. If $H$ has no infinite clique, then \[ \FIN(\widehat{\varphi_Z^G})=\FIN(\varphi^H_{\chi_f}). \] \end{proposition} \begin{proof} The graphs $G$ and $H$ have the same homogeneous sets. Since all cliques of $H$ are finite, its independent sets generate $\mathcal I_G$. The compact hereditary families $\hg(G)$ and $\mathrm{Ind}(H)$ therefore generate the same ideal. By the independence of the generating family in Theorem~\ref{thm:least-extension}, \[ \FIN(\widehat{\varphi_Z^G})=\FIN(\widehat{\varphi_\chi^H}). \] Theorem~\ref{thm:env} identifies $\widehat{\varphi_\chi^H}$ with $\varphi_{\chi_f}^H$. \end{proof} Under these hypotheses, allowing all homogeneous sets in place of independent sets of $H$ may change the fractional covering values, but it does not change which sets have finite value. Thus, the chromatic and cochromatic constructions determine the same least nonpathological $F_\sigma$ extension whenever it is proper. For the random graph, the two constructions do differ at the level of ideals: an infinite clique $C$ satisfies $\varphi_{Z_f}^R(C)=1$, whereas $\varphi_{\chi_f}^R(C)=\infty$. We therefore use the homogeneous generators of $\R$. Write $\varphi_Z=\varphi_Z^R$ and $\varphi_{Z_f}=\varphi_{Z_f}^R$, so $\R=\FIN(\varphi_Z)$. The fact that $\FIN(\varphi_{Z_f})$ is proper follows from the next proposition. \begin{proposition}\label{prop:erdos} For every integer $n\geq1$, there is a set $W_n\subseteq\w$ with $|W_n|=n^2$ and $\varphi_{Z_f}(W_n)\geq n$. In particular, $\varphi_{Z_f}$ is unbounded. \end{proposition} \begin{proof} Fix $n\geq1$. By universality, choose $W_n\subseteq\w$ such that $H=R[W_n]$ is a disjoint union of $n$ copies of $K_n$. Every clique of $H$ is contained in one copy, and every independent set contains at most one vertex from each copy. Thus, every homogeneous set has at most $n$ vertices. For any fractional cocoloring $w$ of $H$, we have $\sum_{J\in\hg(H),\,v\in J}w(J)\geq1$ for every $v\in W_n$. Summing these inequalities over all vertices gives \[ n^2\leq\sum_{J\in\hg(H)}|J|w(J) \leq n\sum_{J\in\hg(H)}w(J). \] Hence, $\varphi_{Z_f}(W_n)=Z_{\mathrm f}(H)\geq n$.\footnote{In fact, $Z_{\mathrm f}(H)=n$: assigning weight $1$ to each of the $n$ cliques gives a fractional cocoloring of total weight $n$.} \end{proof} \begin{theorem}\label{thm:B} The ideal $\FIN(\varphi_{Z_f})$ is the least nonpathological $F_\sigma$ extension of $\R$, and \[ \R\subsetneq\FIN(\varphi_{Z_f})\subsetneq\mathcal P(\w). \] \end{theorem} \begin{proof} By Proposition~\ref{prop:cochromatic-envelope}, $\varphi_{Z_f}=\widehat{\varphi_Z}$, and Proposition~\ref{prop:erdos} shows that this submeasure is unbounded. Hence, $\FIN(\varphi_{Z_f})$ is proper. Theorem~\ref{thm:least-extension}, applied to $\hg(R)$, makes it the least nonpathological $F_\sigma$ extension of $\R$. It remains to prove that $\R\subsetneq\FIN(\varphi_{Z_f})$. The Kneser set $X$ from Corollary~\ref{cor:kneser-witness} satisfies $X\notin\R$.\footnote{Recall that $R[X]$ is the disjoint union of the graphs $R[X_k]\cong\KG(3k-1,k)$, for $k\geq2$.} On the other hand, for every finite $F\subseteq X$, the disjoint-union formula \eqref{eq:frac-disjoint} and the star covers in \eqref{eq:kneser-fractional-bound} give \[ Z_{\mathrm f}(R[F]) \leq\chif(R[F]) =\max_{k\geq2}\chif(R[F\cap X_k]) \leq3. \] The first inequality holds because every independent set is homogeneous. Taking the supremum over finite $F\subseteq X$ gives $\varphi_{Z_f}(X)\leq3$,\footnote{In fact, $\varphi_{Z_f}(X)=3$: by \cite[Example~2, p.~6]{GKM}, $Z_{\mathrm f}(R[X_k])=3-1/k$, so $\varphi_{Z_f}(X)\geq\sup_{k\geq2}(3-1/k)=3$.} so $X\in\FIN(\varphi_{Z_f})\setminus\R$.\footnote{Since $\FIN(\varphi_{Z_f})$ is nonpathological, strictness also follows from the pathology of $\R$ in Corollary~\ref{cor:kneser-witness}. The proof above gives an explicit witness.} \end{proof} One might try to prove pathology of $\R$ by applying Lemma~\ref{lem:blocks} directly to $\varphi_Z$ on all of $\w$. The following fact rules out the uniform covering bounds required for this approach. \begin{fact}[The block criterion on the whole space] Let $(K_n)_{n\in\w}$ be an increasing exhaustion of $\w$ by nonempty finite sets, and let $\mathcal S_n=(S_i^n)_{i0$ such that \[ \delta(K_n,\mathcal S_n)\geq\varepsilon, \qquad \varphi_Z(S_i^n)\leq M \qquad(n\in\w,\ i0$ there is $c>0$ such that every graph on $N$ vertices with no homogeneous set of cardinality at least $a\log_2 N$ contains every graph on at most $c\log_2 N$ vertices as an induced subgraph. \end{theorem} \begin{proposition}\label{prop:erdosseq} Let $(G_n)_{n\in\w}$ be an Erd\H{o}s sequence and put $G=\bigsqcup_{n\in\w}G_n$. Then $\mathcal I_G$ is pathological and has a positive hereditarily pathological restriction. In particular, if $R[X]\cong G$, then $X\in\R^+$ and $\R\res X$ has the same properties. \end{proposition} \begin{proof} We will find an induced copy of the disjoint union of Kneser graphs from Corollary~\ref{cor:kneser-witness}. Let $c>0$ be given by Theorem~\ref{thm:promel-rodl} for $a=4$. For all sufficiently large $n$, every homogeneous set in $G_n$ has size at most $2\log_2|V_n|<4\log_2|V_n|$, so the theorem applies to $G_n$. Since $|V_n|\to\infty$, we can choose strictly increasing indices $n_k$, for $k\geq2$, such that \[ c\log_2|V_{n_k}|\geq |V(\KG(3k-1,k))|. \] Thus, $G_{n_k}$ contains an induced copy of $\KG(3k-1,k)$ on a set $B_k\subseteq V_{n_k}$. Theorem~\ref{thm:lovasz} and equation~\eqref{eq:kneser-fractional-bound} give \[ \chi(G[B_k])=k+1, \qquad \chif(G[B_k])<3. \] Put $A=\bigcup_{k\geq2}B_k$. Since there are no edges between distinct $B_k$, Lemma~\ref{lem:transfer} shows that $\mathcal I_G\res A$ is isomorphic to the Kneser restriction in Corollary~\ref{cor:kneser-witness}. Thus, $A\in\mathcal I_G^+$ and $\mathcal I_G\res A$ is hereditarily pathological. Fact~\ref{fact:restr} then gives pathology of $\mathcal I_G$. Finally, if $X\subseteq\w$ induces a copy of the disjoint union in the random graph, Lemma~\ref{lem:transfer} identifies $\R\res X$ with the same graph block ideal. Since that ideal is proper, the lemma also gives $X\in\R^+$. \end{proof} The Erd\H{o}s sequences show that condition~\eqref{eq:hereditary-bound} in Theorem~\ref{thm:nonpath-char} cannot be restricted to whole blocks. For a nonempty finite graph $H$, write $\alpha(H)$ for the maximum size of an independent set in $H$. Summing the fractional covering inequalities gives $\chif(H)\geq |V(H)|/\alpha(H)$. Since $(G_n)$ is an Erd\H{o}s sequence, for all sufficiently large $n$ we obtain \[ \chif(G_n)\geq\frac{|V_n|}{\alpha(G_n)} \geq\frac{|V_n|}{2\log_2|V_n|} \longrightarrow\infty. \] Hence, for each fixed $C>0$, only finitely many blocks satisfy $\chif(G_n)\leq C$, and their chromatic numbers are bounded. The condition therefore holds when tested only on whole blocks, but fails on the induced subgraphs $G[B_k]$ constructed above, whose fractional chromatic numbers are less than $3$ and whose chromatic numbers are unbounded. We conclude by examining the role of the chosen covers in \cite[Question~3.10]{MMU}. Under the hypotheses of their Theorem~3.9, the authors ask whether boundedness of the envelope implies a positive uniform lower bound on the covering densities, and whether the absence of such a bound implies nonpathology of the ideal. Both questions were answered negatively by Filip\'ow and Tryba using singleton covers \cite[Lemma~11.7 and the discussion following it]{FT}. The Kneser construction also illustrates why the choice of covers matters. Let $G$ be the disjoint union of Kneser graphs used in Corollary~\ref{cor:kneser-witness}, and put $\varphi=\varphi_\chi^G$. Then $\varphi$ is unbounded, $\widehat\varphi(V(G))\leq3$, and $\FIN(\varphi)$ is pathological. Nevertheless, for any increasing exhaustion $(K_n)_{n\in\w}$ of $V(G)$ by nonempty finite sets, the families $\mathcal S_n=\{\{v\}\mid v\in K_n\}$ consist of sets of $\varphi$-value $1$ and satisfy \[ \delta(K_n,\mathcal S_n)=\frac1{|K_n|}\longrightarrow0. \] This suggests considering the best available covering densities. We optimize over finite covering sequences, allowing repetitions, rather than over families of distinct sets. Let $\mathcal C\subseteq2^V$ be compact and hereditary and contain all singletons. For nonempty finite $F\subseteq V$, put\footnote{Recall that $\mathcal C_F=\mathcal C\cap\mathcal P(F)$.} \[ \delta_{\mathcal C}^{*}(F) = \max\left\{ \delta(F,\mathcal S)\mathrel{}\middle|\mathrel{} \mathcal S\text{ is a finite covering sequence of }F \text{ with terms in }\mathcal C_F \right\}. \] Finite covering duality gives \[ \delta_{\mathcal C}^{*}(F) = \frac1{\widehat{\varphi_{\mathcal C}}(F)}. \] Indeed, an optimal fractional cover can be chosen with rational weights. Clearing denominators gives a covering sequence of density at least $1/\widehat{\varphi_{\mathcal C}}(F)$. Conversely, a sequence of length $m$ and density $\delta>0$ gives a fractional cover of total weight $1/\delta$: assign weight $1/(m\delta)$ to each term, adding weights when a set is repeated. Proposition~\ref{prop:cover-duality} gives the reverse inequality and shows that the maximum is attained. Consequently, for every increasing exhaustion $(K_n)_{n\in\w}$ of $V$ by nonempty finite sets, \[ \widehat{\varphi_{\mathcal C}}(V)<\infty \quad\Longleftrightarrow\quad \inf_{n\in\w}\delta_{\mathcal C}^{*}(K_n)>0. \] Thus, for covering submeasures, boundedness of the envelope is equivalent to the existence of covering sequences whose densities have a positive uniform lower bound.\footnote{More generally, let $\varphi$ be an lscsm and let $\mathcal S=(S_i)_{i0$ there is $M_C<\infty$ such that, for every $n\in\omega$ and $B\subseteq V_n$, \[ \chif(G_n[B])\leq C \quad\Longrightarrow\quad \chi(G_n[B])\leq M_C. \] In this case, $\mathcal I_G$ is nonpathological. \item[(b)] The uniform bound in (a) fails and $\sup_{n\in\omega}\chif(G_n)=\infty$. In this case, $\mathcal I_G$ is pathological but not super-pathological, and it has a super-pathological positive restriction. \item[(c)] $\sup_{n\in\omega}\chif(G_n)<\infty$. In this case, $\mathcal I_G$ is super-pathological. \end{enumerate} \end{corollary} \begin{proof} We first check that the three alternatives are mutually exclusive and exhaustive. If (a) holds, the fractional chromatic numbers of the whole blocks must be unbounded: otherwise, taking $B=V_n$ would give a uniform bound on $\chi(G_n)$, contrary to the hypothesis. Thus, neither (b) nor (c) holds. If (a) fails, unbounded fractional chromatic numbers give case (b), while bounded fractional chromatic numbers give case (c). For a disjoint union of finite graphs, condition (a) is equivalent to condition (iii) of Theorem~\ref{thm:nonpath-char}, which gives nonpathology in that case. In case (b), Theorem~\ref{thm:nonpath-char} gives pathology, while Theorem~\ref{thm:C} rules out super-pathology. The existence of a super-pathological positive restriction follows from Theorem~\ref{thm:restriction-extension}. Finally, Theorem~\ref{thm:C} gives super-pathology in case (c). \end{proof} Theorem~\ref{thm:C} also gives a descriptive complexity bound for super-pathology. In the sequence coding of Remark~\ref{rem:complexity}, restrict attention to block sequences with unbounded chromatic numbers. Within this class, super-pathology is expressed by \[ (\exists m\in\omega)(\forall n\in\omega)\;\chif(G_n)\leq m. \] For fixed $m$ and $n$, the inequality depends only on the finite graph $G_n$, so it defines a clopen condition in the sequence space. Thus, super-pathology has a $\Sigma^0_2$ definition relative to this class. For nonpathology, Remark~\ref{rem:complexity} gives a $\Pi^0_3$ upper bound. For finite uniform hypergraphs, define $\chi$ and $\chif$ using independent covers and fractional independent covers, respectively, with both values zero on the empty hypergraph. The classification in Corollary~\ref{cor:tri} also holds for disjoint unions $G=\bigsqcup_{n\in\w}G_n$ of nonempty finite $d$-uniform hypergraphs, where $d\geq2$ and $\sup_{n\in\w}\chi(G_n)=\infty$. To see this, put $V_n=V(G_n)$ and $\mathcal I_G=\langle\hg(G)\rangle$, and let $\varphi_\chi^G$ be the covering submeasure of the compact hereditary family $\mathrm{Ind}(G)$. Then $\mathcal I_G=\FIN(\varphi_\chi^G)$ and, for $A\subseteq V(G)$, \[ \begin{aligned} \varphi_\chi^G(A)&=\sup_{n\in\w}\chi(G_n[A\cap V_n]),\\ \widehat{\varphi_\chi^G}(A)&=\sup_{n\in\w}\chif(G_n[A\cap V_n]). \end{aligned} \] Indeed, every complete set is finite: it either lies in one block or has fewer than $d$ vertices. Replacing complete sets by singletons and combining independent sets from different blocks gives the representation of $\mathcal I_G$ and the formula for $\varphi_\chi^G$. For the envelope, apply Proposition~\ref{prop:cover-duality}. The fractional chromatic number of a finite disjoint union is the maximum of the values of its components: clear a common denominator in optimal rational covers, extend each resulting sequence to the maximum of their lengths by adding empty terms, and unite corresponding terms. Restriction gives the reverse inequality, and lower semicontinuity gives the formula on arbitrary sets. Since $\sup_{n\in\w}\chi(G_n)=\infty$, the ideal $\mathcal I_G$ is proper. By Theorem~\ref{thm:least-extension}, it is nonpathological exactly when $\FIN(\varphi_\chi^G)=\FIN(\widehat{\varphi_\chi^G})$. The uniform bound in Corollary~\ref{cor:tri}(a) implies this equality. If the bound fails for some $C>0$, choose $B_k\subseteq V_{n_k}$ with $\chi(G_{n_k}[B_k])>k$ and $\chif(G_{n_k}[B_k])\leq C$, using distinct blocks. This is possible because the induced chromatic numbers of finitely many blocks are bounded. Then $A=\bigcup_{k\in\w}B_k$ is positive and $\widehat{\varphi_\chi^G}(A)\leq C$. Theorem~\ref{thm:restriction-extension} shows that $\mathcal I_G\res A$ is super-pathological, hence hereditarily pathological. The same theorem, applied to $V(G)$, gives super-pathology of $\mathcal I_G$ exactly when $\sup_{n\in\w}\chif(G_n)<\infty$. These conclusions give the three cases of Corollary~\ref{cor:tri}. We now return to the random ideals $\R_d$, $d\geq2$. The transferred Kneser restrictions are super-pathological, and the positive nonpathological restrictions prevent super-pathology of the ambient ideals. \begin{corollary}\label{cor:random-strong} For every $d\geq2$, $\R_d$ is pathological but not super-pathological, and has both a super-pathological positive restriction and a nonpathological positive restriction isomorphic to $\EDfin$. \end{corollary} \begin{proof} Theorem~\ref{thm:Rd-copy} gives pathology and the nonpathological restriction. The latter rules out super-pathology by Fact~\ref{fact:pos-restr}. By Proposition~\ref{prop:strong-block}, the Kneser restriction of $\R$ is super-pathological. The positive copy of $\R$ supplied by Theorem~\ref{thm:Rd-copy} transfers this restriction to $\R_d$. The canonical envelopes in Theorem~\ref{thm:B} and Proposition~\ref{prop:dprob} also give explicit proper nonpathological extensions of the ambient ideals. \end{proof} Corollary~\ref{cor:random-strong} shows that neither nonpathology nor super-pathology passes from a positive restriction to the ambient ideal. For super-pathology, positivity can be lost when the ideal is enlarged: a set $Y\notin\mathcal I$ may belong to a proper extension $\mathcal J\supseteq\mathcal I$. Then every subset of $Y$ belongs to $\mathcal J$, so $\mathcal J\res Y=\mathcal P(Y)$. Thus, restricting a proper nonpathological extension of $\mathcal I$ need not produce a proper extension of $\mathcal I\res Y$. Indeed, $\FIN(\widehat{\varphi_Z^R})$ contains the Kneser set $X$, by the proof of Theorem~\ref{thm:B}, although $\R\res X$ is super-pathological. Consequently, \[ \FIN(\widehat{\varphi_Z^R})\res X=\mathcal P(X). \] The extension therefore becomes improper on $X$, so it does not contradict super-pathology of $\R\res X$. The random graph also realizes the intermediate case of the block classification. \begin{corollary}\label{cor:threereg} All three cases of Corollary~\ref{cor:tri} occur among the restrictions of $\R$ to positive sets inducing disjoint unions of finite graphs. In particular, if $Y\subseteq\w$ induces a copy of \[ \bigsqcup_{k\geq2}\bigl(K_k\sqcup\KG(3k-1,k)\bigr), \] then $Y\in\R^+$ and $\R\res Y$ is pathological but not super-pathological. \end{corollary} \begin{proof} Fact~\ref{fact:edfin} gives the nonpathological case. The super-pathological Kneser restriction is supplied by Proposition~\ref{prop:strong-block}. To obtain the intermediate case, use each $K_k$ and each $\KG(3k-1,k)$ as a separate block. The two families supply the two requirements of Corollary~\ref{cor:tri}(b). The Kneser blocks have fractional chromatic numbers below $3$ and chromatic numbers $k+1$, by Theorem~\ref{thm:lovasz} and equation~\eqref{eq:kneser-fractional-bound}. Thus, the uniform bound in Corollary~\ref{cor:tri}(a) fails already on these whole blocks. On the other hand, $\chif(K_k)=k$, so the sequence of fractional chromatic numbers is unbounded. Corollary~\ref{cor:tri}(b) therefore gives the intermediate case. Universality supplies an induced copy on a set $Y$, and Lemma~\ref{lem:transfer} gives its positivity and identifies its ideal with $\R\res Y$. \end{proof} The Erd\H{o}s sequences from Section~\ref{sec:global} provide another source of this intermediate behavior. \begin{corollary}\label{cor:erdos-strong} The graph block ideal of every Erd\H{o}s sequence, and of each of its infinite subsequences, is pathological but not super-pathological and has a super-pathological positive restriction. The same holds for induced block copies in $\R$. \end{corollary} \begin{proof} Proposition~\ref{prop:erdosseq} gives pathology and a positive restriction whose block covers satisfy Proposition~\ref{prop:strong-block}. The estimate following Proposition~\ref{prop:erdosseq} shows that the fractional chromatic numbers of the blocks tend to infinity, so Theorem~\ref{thm:C} rules out super-pathology of the graph block ideal. Every infinite subsequence is again an Erd\H{o}s sequence, and Lemma~\ref{lem:transfer} gives the assertion about induced copies. \end{proof} These examples distinguish having a proper summable extension from being an intersection of summable ideals. The ideal $\R$ has a proper summable extension, by Corollaries~\ref{cor:random-strong} and~\ref{cor:summable-intersection}. It is not an intersection of summable ideals, by its pathology and Corollary~\ref{cor:summable-intersection}. The same holds for every ideal in case (b) of Corollary~\ref{cor:tri}. We finally consider the universal hypergraphs obtained by forbidding a finite complete hypergraph. For certain forbidden sizes, their homogeneous-set ideals occur alongside $\R_d$, $d\geq3$, in the super-pathology assertion of \cite[Corollary~5.18]{LOU}. We show that these ideals are pathological but not super-pathological for every forbidden size $n\geq d+1$: the transferred Kneser witness gives positive super-pathological restrictions, and a separate finite construction makes the canonical envelopes unbounded. Remark~\ref{rem:LOU} explains the relation with \cite[Corollary~5.18]{LOU}. For $d\geq3$ and $n\geq d+1$, let $K_n^{(d)}$ be the complete $d$-uniform hypergraph on $n$ vertices, and let $R^d_{n\text{-free}}$ denote the countable universal homogeneous $K_n^{(d)}$-free hypergraph; see \cite[\S5.1]{LOU}. Thus, $R^d_{n\text{-free}}$ contains no complete subset of cardinality $n$ and contains an induced copy of every countable $d$-uniform hypergraph with this property. \begin{proposition}\label{prop:nfree-path} For every $d\geq3$ and $n\geq d+1$, the homogeneous-set ideal of $R^d_{n\text{-free}}$ has a positive super-pathological restriction. In particular, it is pathological. \end{proposition} \begin{proof} Let $G=\bigsqcup_{k\geq2}\KG(3k-1,k)$. This graph has no triangles, since three pairwise disjoint $k$-subsets would require at least $3k$ points. By Fact~\ref{fact:dimension-homogeneity}, $G^{(d)}$ is $K_{d+1}^{(d)}$-free, and Lemma~\ref{lem:dimension-transfer} shows that it has the same homogeneous-set ideal as $G$. That ideal is proper and super-pathological by Proposition~\ref{prop:strong-block}. Since $n\geq d+1$, the hypergraph $G^{(d)}$ is also $K_n^{(d)}$-free. Embed it as an induced subhypergraph of $R^d_{n\text{-free}}$. Fact~\ref{fact:induced-restr} identifies its ideal with a positive restriction of the ambient ideal, and Fact~\ref{fact:restr} gives pathology. \end{proof} The following lemma gives the finite construction needed for Proposition~\ref{prop:nfree-env}. Its proof adapts the probabilistic argument of \cite[Theorem~3.1.2]{AS} from graphs to $d$-uniform hypergraphs. \begin{lemma}\label{lem:finite-nfree} For every integer $d\geq3$ and every integer $m\geq1$, there is a nonempty finite $K_{d+1}^{(d)}$-free $d$-uniform hypergraph $H$ such that every homogeneous set has cardinality less than $|V(H)|/m$. \end{lemma} \begin{proof} Fix $d$ and $m$. For an integer $k\geq2d$, choose each possible edge on $mk$ vertices independently with probability $1/(mk)$. We seek a choice with no complete subset of cardinality $d+1$ and no independent subset of cardinality $k$. Let $Z$ count the complete subsets of cardinality $d+1$. Since each such subset requires $d+1$ edges, its expectation is \[ \mathbb{E}(Z)=\binom{mk}{d+1}\left(\frac{1}{mk}\right)^{d+1}<\frac{1}{(d+1)!}. \] \displaywidowpenalty=10000 Let $U$ count the independent subsets of cardinality $k$. Using $\binom{mk}{k}\leq2^{mk}$ and $\binom{k}{d}\geq\frac{k^d}{2^d d!}$, we obtain \[ \begin{aligned} \mathbb{E}(U) &=\binom{mk}{k}\left(1-\frac{1}{mk}\right)^{\binom{k}{d}}\\ &\leq\exp\left(mk\ln2-\frac{1}{mk}\binom{k}{d}\right)\\ &\leq\exp\left(mk\ln2-\frac{k^{d-1}}{2^d m d!}\right). \end{aligned} \] The last expression tends to zero as $k\to\infty$: since $d\geq3$ and $d,m$ are fixed, the negative term in the exponent grows at least quadratically, whereas the positive term grows linearly. Thus, $\mathbb{E}(Z+U)<1$ for all sufficiently large $k$. Since $Z+U$ is a nonnegative integer, some choice has $Z=U=0$. The resulting hypergraph is $K_{d+1}^{(d)}$-free; its complete sets have cardinality at most $d