On the minimum degree of minimal Ramsey graphs for multiple colours

A graph G is r-Ramsey for a graph H, denoted by G→ (H)r, if every r-colouring of the edges of G contains a monochromatic copy of H. The graph G is called r-Ramseyminimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property. Let sr(H) denote the smallest minimum degree of G over all graphs G that are r-Ramseyminimal for H. The study of the parameter s2 was initiated by Burr, Erdős, and Lovász in 1976 when they showed that for the clique s2(Kk) = (k − 1). In this paper, we study the dependency of sr(Kk) on r and show that, under the condition that k is constant, sr(Kk) = r 2 · polylog r. We also give an upper bound on sr(Kk) which is polynomial in both r and k, and we determine sr(K3) up to a factor of log r.