Topology of unavoidable complexes

Abstract

The partition number $π(K)$ of a simplicial complex $K\subset 2^{[m]}$ is the minimum integer $ν$ such that for each partition $A_1\uplus\ldots\uplus A_ν= [m]$ of $[m]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is $r$-unavoidable if $π(K)\leq r$. We say that a complex $K$ is globally $r$-non-embeddable in $\mathbb{R}^d$ if for each continuous map $f: | K| \rightarrow \mathbb{R}^d$ there exist $r$ vertex disjoint faces $σ_1,\ldots, σ_r$ of $| K|$ such that $f(σ_1)\cap\ldots\cap f(σ_r)\neq\emptyset$. Motivated by the problems of Tverberg-Van Kampen-Flores type we prove several results (Theorems 3.6, 3.9, 4.6) which link together the combinatorics and topology of these two classes of complexes. One of our central observations (Theorem 4.6), summarizing and extending results of G. Schild, B. Grünbaum and many others, is that interesting examples of (globally) $r$-non-embeddable complexes can be found among the joins $K = K_1\ast\ldots\ast K_s$ of $r$-unavoidable complexes.