论文标题
用于彩色的拉姆西编号的下限
A lower bound for set-colouring Ramsey numbers
论文作者
论文摘要
固定颜色的Ramsey编号$ r_ {r，s}（k）$被定义为最低$ n $，这样，如果将完整图的每个边缘分配给$ k_n $的每个边缘，则从$ \ {1，\ ldots，r \} $中分配了一组$ s $颜色，其中一种颜色包含单色$ k $的单色$ k $。案例$ s = 1 $是通常的$ r $ - 彩色拉姆齐号，$ s = r -1 $由Erdős，Hajnal和Rado在1965年研究，以及1972年的Erdős和Szemerédi。 $ s $的第一个重要结果仅是最近才由Conlon，Fox，He，Mubayi，Suk和Verstraëte获得的，他们表明$ R_ {R，S}（K）= 2^{θ（KR）} $，如果$ S/R $从$ 0 $ 0 $ 0 $和$ 1 $中限制。但是，在$ s = r -o（r）$的范围内，它们的上限和下限显着差异。在本说明中，我们引入了一种新的（随机）着色，并使用它来确定$ r_ {r，s}（k）$ up up to the exponent中的polygarithmic因素，对于所有$ r $，$ s $和$ k $。
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 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 $R_{r,s}(k) = 2^{Θ(kr)}$ 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 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 $k$.
