Open Collections

Description

Pach's selection theorem asserts that for any positive integer $d$ there exists a constant $c_d > 0$ such that for any positive integer $n$ and any finite sets $X_1, ..., X_{d+1} in \mathbb{R}^d$ each with $n$ points there exist disjoint subsets $Z_1, ..., Z_{d+1}, Z_i$ is a subset of $X_i$ and a point $z$ such that $z$ belongs to any rainbow $(Z_1, ..., Z_{d+1})$-simplex; that is, a convex hull of points $z_1, ..., z_{d+1}$ where $z_i$ belongs to $Z_i$. Although the topological method is a valuable tool for improving the bounds for certain selection theorems (introduced by Gromov), we prove that Pach's theorem does not admit a topological extension. Joint work with Imre B\'ar\'any, Roy Meshulam and Eran Nevo.