An r-partite hole of size k in an r-uniform hypergraph H is a collection of pairwise disjoint vertex subsets V_1, ..., V_r, all of size k such that no edge touches each of V_1,..., V_r. Let a_r(H) be the largest size of an r-partite hole in H. We determine a relationship between a_r(H) and the order of the largest monochromatic component in an arbitrary edge coloring of H. We discuss some implications for random graphs and hypergraphs as well as random Steiner triple systems.
Joint work with Louis DeBiasio.
