Pach's selection theorem does not admit a topological extension

TL;DR

This paper investigates the possibility of extending Pach's selection theorem topologically and finds that a linear-size extension is impossible, but a logarithmic-size extension is achievable.

Contribution

It proves the non-existence of a topological extension with linear-size sets and constructs a topological extension with logarithmic-size sets.

Findings

Linear-size topological extension is impossible.

A topological extension with size $( ext{log} n)^{1/d}$ exists.

Abstract

Let $U_1,\dots, U_{d+1}$ be $n$-element sets in $R^d$ and let $\langle u_1,\ldots,u_{d+1}\rangle$ denote the convex hull of points $u_i$ in $U_i$ (for all $i$) which is a (possibly degenerate) simplex. Pach's selection theorem says that there are sets $Z_1 \subset U_1,\dots, Z_{d+1} \subset U_{d+1}$ and a point $u$ in $R^d$ such that each $|Z_i| > c_1(d)n$ and $u$ belongs to $\langle z_1,...,z_{d+1} \rangle$ for every choice of $z_1$ in $Z_1,\dots,z_{d+1}$ in $Z_{d+1}$. Here we show that this theorem does not admit a topological extension with linear size sets $Z_i$. However, there is a topological extension where each $|Z_i|$ is of order $(\log n)^(1/d)$.