For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is a subset of $n$ vertices where all of the edges between them receive a common color. If $n$ is fixed and $\frac{s}{r}$ is less than and bounded away from $1-\frac{1}{n-1}$, then $R(n;r,s)$ is known to grow exponentially in $r$, while if $\frac{s}{r}$ is greater than and bounded away from $1-\frac{1}{n-1}$, then $R(n;r,s)$ is bounded. Here we prove bounds for $R(n;r,s)$ in the intermediate range where $\frac{s}{r}$ is close to $1 - \frac{1}{n-1}$ by establishing a connection to the maximum size of error-correcting codes near the zero-rate threshold.

翻译：对于满足$r > s$的正整数$n,r,s$，集合着色拉姆齐数$R(n;r,s)$定义为最小的$N$，使得若完全图$K_N$的每条边从包含$r$种颜色的调色板中获得$s$种颜色的集合，则存在一个$n$个顶点的子集，其中所有边均接收共同的颜色。当$n$固定且$\frac{s}{r}$小于并远离$1-\frac{1}{n-1}$时，已知$R(n;r,s)$随$r$呈指数增长；而当$\frac{s}{r}$大于并远离$1-\frac{1}{n-1}$时，$R(n;r,s)$有界。本文通过建立与零率阈值附近最大纠错码规模的联系，证明了$\frac{s}{r}$接近$1 - \frac{1}{n-1}$这一中间范围内$R(n;r,s)$的界限。