Topology of unavoidable complexes

Abstract

The partition number π(K) of a simplicial complex K ⊆ 2 [m] is the minimum integer ν such that for each partitionMotivated by the problems of Tverberg-Van Kampen-Flores type we prove several results (Theorems 3.14, 3.18, 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 (almost) r-non-embeddable complexes can be found among the joins K = K 1 * . . . * K s of r-unavoidable complexes.