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 $n, r, s$，集合染色拉姆齐数 $R(n;r,s)$ 定义为最小的 $N$，使得若完全图 $K_N$ 的每条边都从包含 $r$ 种颜色的调色板中接收 $s$ 种颜色，则存在一个由 $n$ 个顶点构成的子集，使得该子集中所有边均接收一种公共颜色。当 $n$ 固定且 $\frac{s}{r}$ 小于并远离 $1-\frac{1}{n-1}$ 时，已知 $R(n;r,s)$ 关于 $r$ 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纠错码最大规模的联系，证明了 $\frac{s}{r}$ 接近 $1 - \frac{1}{n-1}$ 这一中间区间内 $R(n;r,s)$ 的界。