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 $[n]$ 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. 
Marija Jeli\'c Milutinovi\'c, Du\v{s}ko Joji\'c, Marinko   Timotijevi\'c, Sini\v{s}a T. Vre\'cica, Rade T. \v{Z}ivaljevi\'c
math.CO