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DOI: 10.4230/LIPIcs.SoCG.2026.9

Intersection Patterns of Set Systems on Manifolds with Slowly Growing Homological Shatter Functions

Authors: Sergey Avvakumov, Marguerite Bin, and Xavier Goaoc

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract

A theorem of Matoušek asserts that for any k ≥ 2, any set system whose shatter function is o(n^k) enjoys a fractional Helly theorem of order k: in the k-wise intersection hypergraph, positive density implies a linear-size clique. Kalai and Meshulam conjectured a generalization of that phenomenon to homological shatter functions. It was verified for set systems with bounded homological shatter functions and whose ground set has a forbidden homological minor (which includes ℝ^d by a homological analogue of the van Kampen-Flores theorem). We present two contributions to this line of research: - We study homological minors in certain manifolds (possibly with boundary), for which we prove analogues of the van Kampen-Flores theorem and of the Hanani-Tutte theorem. - We introduce graded analogues of the Radon and Helly numbers of set systems and relate their growth rate to the original parameters. This allows to extend the verification of the Kalai-Meshulam conjecture to sufficiently slowly growing homological shatter functions.

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Sergey Avvakumov, Marguerite Bin, and Xavier Goaoc. Intersection Patterns of Set Systems on Manifolds with Slowly Growing Homological Shatter Functions. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{avvakumov_et_al:LIPIcs.SoCG.2026.9,
  author =	{Avvakumov, Sergey and Bin, Marguerite and Goaoc, Xavier},
  title =	{{Intersection Patterns of Set Systems on Manifolds with Slowly Growing Homological Shatter Functions}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{9:1--9:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.9},
  URN =		{urn:nbn:de:0030-drops-258152},
  doi =		{10.4230/LIPIcs.SoCG.2026.9},
  annote =	{Keywords: Fractional Helly theorem, homological minor, combinatorial convexity}
}

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Intersection Patterns of Set Systems on Manifolds with Slowly Growing Homological Shatter Functions

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