# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

__Mittagsseminar Talk Information__ | |

**Date and Time**: Tuesday, June 10, 2014, 12:15 pm

**Duration**: 30 minutes

**Location**: OAT S15/S16/S17

**Speaker**: Zuzana Safernová (Charles University)

## Bounds for Pach's selection theorem

We focus on the estimates on the selection constant in the following geometric selection theorem by Pach: For every positive integer *d* there is a constant *c*_{d} > 0 such that whenever *X*_{1}, … , X_{d+1} are *n*-element subsets of **R**^{d}, then we can find a point *p* in **R**^{d} and subsets *Y*_{i} ⊂ X_{i} for every *i*=1,…, *d*+1, each of size at least *c*_{d} n, such that *p* belongs to all *rainbow* *d*-simplices determined by *Y*_{1}, … , Y_{d+1}, that is, simplices with one vertex in each *Y*_{i}.

We will discuss a lower and upper bound for the constant *c*_{d}, more concretely, *2*^{-2^{d^2 + O(d)}} ≤ c_{d} ≤ 0.9975^{d}. In order to show the upper bound, we prove the fact that the minimum solid angle of every *d*-simplex is exponentially small.

Joint work with Honza Kynčl, Pavel Paták and Martin Tancer.

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