Combinatorics of unavoidable complexes

Abstract

The partition number π(K) of a simplicial complex K ⊆ 2 [n] is the minimum integer k such that for each partition A 1 . .Motivated by the Van Kampen-Flores and Tverberg type results, and inspired by the 'constraint method' [BFZ], we study the combinatorics of r-unavoidable complexes. Emphasizing the interplay of ideas from combinatorial topology, linear programming and fractional graph theory, we explore and compare extremal properties of examples arising in topology (minimal triangulations) and combinatorics (hypergraph theory and Ramsey theory).