2022/12/13 by Aragão, Lucas, Collares, Maurício, Marciano, João Pedro +2
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2212.06802
The set-colouring Ramsey number Rr,s(k) is defined to be the minimum n such that if each edge of the complete graph Kn is assigned a set of s colours from \1,…,r\, then one of the colours contains a monochromatic clique of size k. The case s = 1 is the usual r-colour Ramsey number, and the case s = r - 1 was studied by Erdős, Hajnal and Rado in 1965, and by Erdős and Szemerédi in 1972. The first significant results for general s were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstraëte, who showed that Rr,s(k) = 2Θ(kr) if s/r is bounded away from 0 and 1. In the range s = r - o(r), however, their upper and lower bounds diverge significantly. In this note we introduce a new (random) colouring, and use it to determine Rr,s(k) up to polylogarithmic factors in the exponent for essentially all r, s and k.