  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 $[n]$ at least one of the sets $A_i$ is in $K$. A complex $K$
is $r$-unavoidable if $\pi(K)\leq r$. 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.
