# CoSP ZOOM Seminar on topological combinatorics

Starting summer 2020 Shira Zerbib is coordinating a CoSP Zoom seminar on topological combinatorics, replacing a conference on the same topic that was supposed to take place in Prague this summer and was cancelled due to COVID-19.

The talks will be given on Tuesdays and Thursdays at 8:00am (CDT), as of June 30, 2020.

Schedule:

September 17

Denys Bulavka (Charles University):

Optimal bounds for the colorful fractional Helly theorem

The well known fractional Helly theorem and colorful Helly theorem can be merged into so called colorful fractional Helly theorem. It states: For every alpha in (0, 1] and every non-negative integer d, there is beta = beta(alpha, d) in (0, 1] with the following property. Let F_1, ..., F_{d+1} be finite nonempty families of convex sets in R^d of sizes n_1, ..., n_{d+1} respectively. If at least alpha n_1 n_2 ... n_{d+1} of the colorful (d+1)-tuples have a nonempty intersection, then there is i in [d+1] such that F_i contains a subfamily of size at least b n_i. (A colorful (d+1)-tuple is a (d+1)-tuple (C_1,...,C_{d+1}) such that C_i belongs to F_i for every i.)

The colorful fractional Helly theorem was first stated and proved by Bárány, Fodor, Montejano, Oliveros, and Pór in 2014 with beta = alpha/(d+1).

In 2017 Kim proved the theorem with better function beta, which in particular tends to 1 when alpha tends to 1. Kim also conjectured what is the optimal bound for beta(alpha, d) and provided the upper bound example for the optimal bound. The conjectured bound coincides with the optimal bounds for the (non-colorful) fractional Helly theorem proved independently by Eckhoff and Kalai around 1984.

We prove Kim's conjecture by extending Kalai's approach to the colorful scenario. Moreover, we obtain bounds for other collections of sets, not just colorful (d+1)-tuples.

This is a joint work with Afshin Goodarzi and Martin Tancer.

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Summer 2020 seminars:

6/30 - Andeas Holmsen (KAIST)

Title: The Fractional Helly property and topological complexity

Abstract: The fractional Helly theorem is a simple yet remarkable generalization of Helly’s classical theorem on the intersection of convex sets, and it is of considerable interest to extend the fractional Helly theorem beyond the setting of convexity. In this talk I will discuss a recent result which shows that the fractional Helly theorem holds for families of subsets of R^d which satisfy only very weak topological assumptions. This is joint work with Xavier Goaoc and Zuzana Patáková.

7/2 - Florian Frick (Carnegie Mellon University)

Title: The topological Tverberg problem beyond prime powers

Abstract: Given d and q the topological Tverberg problem asks for the minimal n such that any continuous map from the n-dimensional simplex to R^d identifies q points from pairwise disjoint faces. For q a prime power n is (q-1)(d+1). The lower bound follows from a general position argument, the upper bound from equivariant topological methods. It was shown recently that for q with at least two distinct prime divisors the lower bound may be improved. For those q non-trivial upper bounds had been elusive. I will show that n is at most q(d+1)-1 for all q. I had previously conjectured this to be optimal unless q is a prime power. This is joint work with Pablo Soberón.

7/7 - Minki Kim (Technion)

7/9 - Zilin Jiang (MIT)

7/14 - Gabor Simonyi (Renyi Institute)

7/16 - Jinha Kim (Technion)

7/21 - Tomas Kaiser (University of West Bohemia) & Matej Stehlik (Universite Grenoble Alpes) - double talk

7/23 - Amzi Jeffs (University of Washington)

7/28 - Martin Loebl (Charles University)

7/30 - Pablo Soberon (Baruch College)