Combinatorics of `unavoidable complexes'

Abstract

The partition number $\pi (K)$ of a simplicial complex $K\subset 2^{[n]}$ is the minimum integer $\nu$ such that for each partition $A_1\uplus \dots \uplus A_{\nu }=[n]$ of $[n]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is $r$-unavoidable if $\pi (K)\le r$. Motivated by the problems of Tverberg-Van Kampen-Flores type, and inspired by the `constraint method' of Blagojević, Frick, and Ziegler, arXiv:1401.0690 [math.CO], we study the combinatorics of $r$-unavoidable complexes.

This paper is an offspring of the unpublished preprint arXiv:1603.08472 [math.AT]