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-Ramsey-minimal for H if it is r-Ramsey for H but no proper subgraph of G possesses this property. Let s_r(H) denote the smallest minimum degree of G over all graphs G that are r-Ramsey-minimal for H. The study of the parameter s₂ was initiated by Burr, Erdos, and Lovász in 1976 when they showed that for the clique s₂(K_k)=(k-1)². In this paper, we study the dependency of s_r(K_k) on r and show that, under the condition that k is constant, s_r(K_k)=r²•polylog r. We also give an upper bound on s_r(K_k) which is polynomial in both r and k, and we show that cr²ln r ≤ s_r(K₃) ≤ Cr²ln²r for some constants c, C>0.