eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-06-02
40:1
40:17
10.4230/LIPIcs.SoCG.2021.40
article
A Stepping-Up Lemma for Topological Set Systems
Goaoc, Xavier
1
Holmsen, Andreas F.
2
Patáková, Zuzana
3
LORIA, Université de Lorraine, France
Department of Mathematical Sciences, KAIST, Daejeon, South Korea
Department of Algebra, Faculty of Mathematics and Physics, Charles University, Czech Republic
Intersection patterns of convex sets in ℝ^d have the remarkable property that for d+1 ≤ k ≤ 𝓁, in any sufficiently large family of convex sets in ℝ^d, 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 ℝ^d. 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol189-socg2021/LIPIcs.SoCG.2021.40/LIPIcs.SoCG.2021.40.pdf
Helly-type theorem
Topological combinatorics
Homological minors
Stair convexity
Cubical complexes
Homological VC dimension
Ramsey-type theorem