2018 May 1

Consider the collection of all the simplices spanned by some n-point set in **R**^{d}. 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_{1}, ..., Q_{d+1}, in **R**^{d} contain linearly-sized subsets P_{i} in Q_{i} 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)^{1/(d-1)}. We show that this is tight in dimension 2, for all surfaces other than S^{2}. Surprisingly, the optimal bound for S^{2} is (log n)^{1/2}. This
improves upon results of Barany, Meshulam, Nevo, Tancer.

To the Combinatorics Seminar Web page.