Intersection patterns of set systems on manifolds with slowly growing homological shatter functions

TL;DR

This paper extends the understanding of intersection patterns in set systems on manifolds with slowly growing homological shatter functions, proving analogues of classical theorems and verifying a conjecture in this context.

Contribution

It introduces graded Radon and Helly numbers, proves analogues of van Kampen-Flores and Hanani-Tutte theorems for manifolds, and verifies the Kalai-Meshulam conjecture under new conditions.

Findings

Proved analogues of van Kampen-Flores theorem for certain manifolds.

Introduced graded Radon and Helly numbers and linked their growth to set system parameters.

Extended verification of the Kalai-Meshulam conjecture for slowly growing homological shatter functions.

Abstract

A theorem of Matou\v{s}ek asserts that for any $k \ge 2$, 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 ground set with a forbidden homological minor (which includes $\mathbb{R}^d$ 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 for sufficiently slowly growing homological shatter functions.