The set-coloring 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 $s$ colors from $\{1,\dots ,r\}$, then one of the colors contains a monochromatic clique of size $k$. The case $s=1$ is the usual $r$-color 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 $R_{r,s}(k)=2^{\mathrm{\Theta }(kr)}$ 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) coloring, and use it to determine $R_{r,s}(k)$ up to polylogarithmic factors in the exponent for essentially all $r$, $s$, and $k$.

中文翻译：

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设定着色拉姆齐数的下界

设定着色拉姆齐数${右}_{r,s}（k）$被定义为最小值$n$这样如果完整图的每条边$K_n$被分配了一组$s$颜色来自$\{1,\mathrm{\dots \dots },r\}$，那么其中一种颜色包含大小为 的单色团$k$。案子$s=1$是平常的$r$- 颜色拉姆齐数和情况$s=r-1$Erdős、Hajnal 和 Rado 在 1965 年进行了研究，Erdős 和 Szemerédi 在 1972 年进行了研究。$s$Conlon、Fox、He、Mubayi、Suk 和 Verstraëte 最近才获得这些结果，他们表明${右}_{r,s}（k）=2^{\mathrm{\theta }（kr）}$如果$s/r$远离 0 和 1。在范围内$s=r-哦（r）$然而，它们的上限和下限存在显着差异。在本笔记中，我们引入了一种新的（随机）着色，并用它来确定${右}_{r,s}（k）$基本上所有指数的多对数因子$r$,$s$， 和$k$。