A Stepping-Up Lemma for Topological Set Systems

Abstract

Intersection patterns of convex sets in R have the remarkable property that for d+1 ≤ k ≤ l, in any sufficiently large family of convex sets in R, 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. Quantitatively, our bounds depend on how complicated the intersection of l elements of F can be, as measured by the maximum of the ⌈ d2 ⌉ 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. 2012 ACM Subject Classification Theory of computation → Computational geometry