On the minimum degree of minimal Ramsey graphs for multiple colours

TL;DR

This paper investigates the minimum degree of r-Ramsey-minimal graphs for cliques, revealing that for fixed k, s_r(K_k) grows approximately as r^2 times a polylogarithmic factor, and provides bounds for these parameters.

Contribution

It establishes the asymptotic behavior of s_r(K_k) for fixed k and explores bounds for the minimum degree in r-Ramsey-minimal graphs for cliques.

Findings

s_r(K_k) = r^2 polylog r for fixed k

Provided polynomial bounds in r and k for s_r(K_k)

Determined s_r(K_3) up to a logarithmic factor

Abstract

A graph G is r-Ramsey for a graph H, denoted by G\rightarrow (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, Erd\H{o}s, and Lov\'{a}sz 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 polylog r. We also give an upper bound on s_r(K_k) which is polynomial in both r and k, and we determine s_r(K_3) up to a factor of log r.