Optimal bounds for the colored Tverberg problem

TL;DR

This paper establishes optimal bounds for the colored Tverberg problem, extending classical theorems with color constraints and providing tight bounds in prime cases using advanced topological methods.

Contribution

It introduces a new intersection theorem with color constraints, improving bounds for the colored Tverberg problem and employing relative equivariant obstruction theory.

Findings

Proves a strengthened Tverberg type intersection theorem with color constraints.

Provides tight bounds for the prime case of the colored Tverberg problem.

Uses topological methods to achieve asymptotically optimal bounds.

Abstract

We prove a "Tverberg type" multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Barany et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Barany & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.