Pach's Selection Theorem Does Not Admit a Topological Extension

Abstract

Let be n-element sets in. Pach's selection theorem says that there exist subsets and a point such that each and for every choice of.

Here we show that this theorem does not admit a topological extension with linear size sets. However, there is a topological extension where each is of order (log n)(1/d).

Keywords

• Pach's Selection Theorem
• Gromov's Overlap Theorem