On the minimum degree of minimal Ramsey graphs

A graph G is r-Ramsey for a graph H, denoted by G ź ( 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 s 2 was initiated by Burr, Erdźs, and Lovasz in 1976 when they showed that for the clique s 2 ( 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 ) = r 2 ź 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 c r 2 ln ź r ≤ s r ( K 3 ) ≤ C r 2 ln 2 ź r for some constants c , C 0 .

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