A stepping-up lemma for topological set systems

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Intersection patterns of convex sets in ℝd have the remarkable property that for d+ 1 ≤ k ≤ ℓ, in any sufficiently large family of convex sets in ℝd, if a constant fraction of the k-element subfamilies have nonempty intersection, then a constant fraction of the ℓ-element subfamilies must also have nonempty intersection. Here, we prove that a similar phenomenon holds for any topological set system F in ℝd. Quantitatively, our bounds depend on how complicated the intersection of ℓ elements of F can be, as measured by the maximum of the ⌈d/2⌉ first Betti numbers. As an application, we improve the fractional Helly number of set systems with bounded topological complexity due to the third author, from a Ramsey number down to d + 1. We also shed some light on a conjecture of Kalai and Meshulam on intersection patterns of sets with bounded homological VC dimension. A key ingredient in our proof is the use of the stair convexity of Bukh, Matoušek and Nivasch to recast a simplicial complex as a homological minor of a cubical complex. © Xavier Goaoc, Andreas F. Holmsen, and Zuzana Patáková; licensed under Creative Commons License CC-BY 4.0 37th International Symposium on Computational Geometry (SoCG 2021).
Publisher
Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Issue Date
2021-06
Language
English
Citation

37th International Symposium on Computational Geometry, SoCG 2021

ISSN
1868-8969
DOI
10.4230/LIPIcs.SoCG.2021.40
URI
http://hdl.handle.net/10203/288772
Appears in Collection
MA-Conference Papers(학술회의논문)
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