Combinatorics of `unavoidable complexes'

The partition number $\pi (K)$ of a simplicial complex $K\subset 2^{[n]}$ is the minimum integer $\nu$ such that for each partition $A_1\uplus\ldots\uplus A_\nu = [n]$of$[n]$atleastoneofthesets$A_i$isin$K$\mathrm{.}Acomplex$K$is$r$-unavoidableif$\pi(K)\leq r$. Motivated by the problems of Tverberg-Van Kampen-Flores type, and inspired by the `constraint method’ of Blagojevi'{c}, Frick, and Ziegler, arXiv:1401.0690 [math.CO], we study the combinatorics of $r$-unavoidable complexes.