Título: A Lower bound for set-colouring Ramsey numbers

Palestrante:Taísa Lopes Martins (IME-UFF)
Data: 19/06/2023
Horário: 15:30 às 16:30h
Local: C116 - Bloco C - CT – Instituto de Matemática – UFRJ.

Resumo: 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) = 2^{Θ(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 talk 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.

This is a joint work with Lucas Aragão, Maurício Collares, João Pedro Marciano and Rob Morris.

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