Colored Tverberg Theorems for non-prime powers

TL;DR

This paper extends the Colored Tverberg theorem to cases where the number of rainbow faces is one less than a prime power, and allows for variable sizes of rainbow simplices, broadening the theorem's applicability.

Contribution

It introduces a generalized version of the Colored Tverberg theorem applicable to non-prime power cases with variable simplex sizes.

Findings

Valid for q = p^n - 1 where p is prime

Allows rainbow simplex sizes of 2q-2 or 2q+1

Generalizes previous theorem to non-prime power scenarios

Abstract

We prove a relative of the Optimal (Type B)} Colored Tverberg theorem of \v{Z}ivaljevi\'{c} and Vre\'{c}ica which modifies this results in two different ways. (1) Our result is valid if the number of rainbow faces is $q= p^n-1$, where $p$ is a prime. (2) The size of rainbow simplices satisfies the condition $\vert C_i\vert \in \{2q-2, 2q+1\}$ while in the original theorem $\vert C_i\vert = 2q-1$ for all $i$.