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- W4311559375 abstract "The set-colouring Ramsey number $R_{r,s}(k)$ is defined to be the minimum $n$ such that if each edge of the complete graph $K_n$ is assigned a set of $s$ colours from ${1,ldots,r}$, 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 ErdH{o}s, Hajnal and Rado in 1965, and by ErdH{o}s and Szemer'edi in 1972. The first significant results for general $s$ were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstraete, who showed that $R_{r,s}(k) = 2^{Theta(kr)}$ 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 $R_{r,s}(k)$ up to polylogarithmic factors in the exponent for essentially all $r$, $s$ and $k$." @default.
- W4311559375 created "2022-12-27" @default.
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- W4311559375 date "2022-12-13" @default.
- W4311559375 modified "2023-09-26" @default.
- W4311559375 title "A lower bound for set-colouring Ramsey numbers" @default.
- W4311559375 doi "https://doi.org/10.48550/arxiv.2212.06802" @default.
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