Tolerance for colorful Tverberg partitions

TL;DR

This paper investigates colorful Tverberg partitions, establishing bounds on the number of color classes needed for convex hulls to intersect despite removal of certain colors, with optimal bounds for specific parameters.

Contribution

It provides asymptotically optimal bounds for colorful Tverberg partitions under color removal, improving previous bounds and characterizing configurations with high tolerance.

Findings

Established asymptotically optimal bounds for $t$ when $r \,\le\, d+1$

Improved bounds for cases when $r > d+1$

Characterized configurations with high tolerance for color removal

Abstract

Tverberg's theorem bounds the number of points $\mathbb{R}^d$ needed for the existence of a partition into $r$ parts whose convex hulls intersect. If the points are colored with $N$ colors, we seek partitions where each part has at most one point of each color. In this manuscript, we bound the number of color classes needed for the existence of partitions where the convex hulls of the parts intersect even after any set of $t$ colors is removed. We prove asymptotically optimal bounds for $t$ when $r \le d+1$, improve known bounds when $r>d+1$, and give a geometric characterization for the configurations of points for which $t=N-o(N)$.