On a topological version of Pach's overlap theorem

TL;DR

This paper investigates a topological extension of Pach's overlap theorem, demonstrating tight bounds in certain surfaces and proposing conjectures for higher-dimensional manifolds, thus advancing understanding of topological intersection properties.

Contribution

The paper shows that a topological version of Pach's theorem does not hold with certain subset sizes, identifies tight bounds for surfaces other than the sphere, and proposes conjectures for higher dimensions.

Findings

Topological extension of Pach's theorem fails with subsets of size C(log n)^{1/(d-1)}.

Tight bounds are established for surfaces other than S^2.

The optimal bound for S^2 in the topological version is of order (log n)^{1/2}.

Abstract

Pach showed that every $d+1$ sets of points $Q_1,\dotsc,Q_{d+1} \subset \mathbb{R}^d$ contain linearly-sized subsets $P_i\subset Q_i$ such that all the transversal simplices that they span intersect. We show, by means of an example, that a topological extension of Pach's theorem does not hold with subsets of size $C(\log n)^{1/(d-1)}$. We show that this is tight in dimension $2$, for all surfaces other than $\mathbb{S}^2$. Surprisingly, the optimal bound for $\mathbb{S}^2$ in the topological version of Pach's theorem is of the order $(\log n)^{1/2}$. We conjecture that, among higher-dimensional manifolds, spheres are similarly distinguished. This improves upon the results of B\'ar\'any, Meshulam, Nevo and Tancer.