HEXAGONbeta

Relative PP-Ideals of Hypergraphs and Ultrafilter Spectra

A mix of human-written and AI-generated textHuman understanding: all partsmath.LO — Logicmath.CO — Combinatoricsmath.GN — General Topology

Contributed by Zhenyu Sun ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

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 PP-property exactly when the filter is the intersection of finitely many PP-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 σ\sigma-ideals exactly for PP-filters, and their uniformities are max⁡{πχ(F),d}\max\{\pi\chi(\mathcal{F}),\mathfrak{d}\}. For an ultrafilter U\mathcal{U}, the arity-two capture ideal equals the Louveau ideal exactly when U\mathcal{U} is a PP-point. Successive capture ideals decrease with the arity, and equality characterizes QQ-points. Every strict inclusion is witnessed by a closed set. Thus a non-QQ PP-point yields a decreasing sequence of σ\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 PP-sets have the same singleton endpoint spectrum and different ray transversal numbers.

Provenance statement

ChatGPT was used for proofreading and language polishing.

Tools used

OpenAI
ChatGPT

References

  1. A. Blass, Combinatorial cardinal characteristics of the continuum , in Handbook of Set Theory (M. Foreman and A. Kanamori, eds.), Springer, Dordrecht, 2010, pp. 395--489. https://doi.org/10.1007/978-1-4020-5764-9_7DOI
  2. J. Brendle and S. Shelah, Ultrafilters on ω\omega---their ideals and their cardinal characteristics , Trans. Amer. Math. Soc. 351 (1999), 2643--2674. https://doi.org/10.1090/S0002-9947-99-02257-6DOI
  3. M. Hrušák, D. Meza-Alcántara, and H. Minami, Pair-splitting, pair-reaping and cardinal invariants of FσF_\sigma-ideals , J. Symbolic Logic 75 (2010), 661--677. https://doi.org/10.2178/jsl/1268917498DOI
  4. S. Koppelberg, 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.
  5. A. Louveau, Une méthode topologique pour l'étude de la propriété de Ramsey , Israel J. Math. 23 (1976), 97--116. https://doi.org/10.1007/BF02756789DOI
  6. O. V. Petrenko and I. V. Protasov, Ultrafilters on GG-spaces , Algebra Discrete Math. 19 (2015), 254--269.
  7. D. Raghavan and S. Shelah, A small ultrafilter number at smaller cardinals , Arch. Math. Logic 59 (2020), 325--334. https://doi.org/10.1007/s00153-019-00693-8DOI
  8. J. Schilhan, Universal domination and idealized forcing , preprint, 2026, arXiv:2608.18964v1 (https://arxiv.org/abs/2608.18964v1).arXiv
  9. P. Vojtáš, Cardinalities of noncentered systems of subsets of ω\omega , Discrete Math. 108 (1992), 125--129. https://doi.org/10.1016/0012-365X(92)90666-4DOI

Version history

  1. v1Submitted by Zhenyu SunInitial depositCurrentOct 20, 2026