On Multicolour Ramsey Numbers and Subset-Colouring of Hypergraphs

For n≥ s> r≥ 1 and k≥ 2, write n→(s)kr if every k-colouring of all r-subsets of an n-element set has a monochromatic subset of size s. Improving upon previous results by Axenovich et al. (Discrete Mathematics, 2014) and Erdős et al. (Combinatorial set theory, 1984) we show that ifr≥3andn↛(s)krthen2n↛(s+1)k+3r+1. This yields an improvement for some of the known lower bounds on multicolour hypergraph Ramsey numbers. Given a hypergraph H= (V, E), we consider the Ramsey-like problem of colouring all r-subsets of V such that no hyperedge of size ≥ r+ 1 is monochromatic. We give upper and lower bounds on the number of colours necessary in terms of the chromatic number χ(H). We show that this number is O(log^(r-1)(rχ(H) ) + r).