====== Boris Bukh (Carnegie Mellon) ======

====== Topological Version of Pach's Overlap Theorem ======

===== Abstract for the Combinatorics and Geometry/Topology Seminars 2018 May 1 =====

Consider the collection of all the simplices spanned by some n-point set in **R**<sup>d</sup>. There are several results showing that simplices defined in this way must overlap very much. In this talk I focus on the generalization of these results to 'curvy' simplices.

Specifically, Pach showed that every d+1 sets of points, Q<sub>1</sub>, ..., Q<sub>d+1</sub>, in **R**<sup>d</sup> contain linearly-sized subsets P<sub>i</sub> in Q<sub>i</sub> such that all the transversal simplices that they span intersect. In joint work with Alfredo Hubard, 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)<sup>1/(d-1)</sup>. We show that this is tight in dimension 2, for all surfaces other than S<sup>2</sup>. Surprisingly, the optimal bound for S<sup>2</sup> is (log n)<sup>1/2</sup>. This improves upon results of Barany, Meshulam, Nevo, Tancer.


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