Set-coloring Ramsey numbers and error-correcting codes near the zero-rate threshold

TL;DR

This paper investigates the growth of set-coloring Ramsey numbers near a critical ratio of colors, establishing bounds by linking them to error-correcting codes close to the zero-rate threshold.

Contribution

It provides new bounds for set-coloring Ramsey numbers in the intermediate regime by connecting them to error-correcting codes near the zero-rate threshold.

Findings

Bounds for $R(n;r,s)$ near the critical ratio are established.

Connection between Ramsey numbers and error-correcting codes is demonstrated.

Results clarify the growth behavior of these numbers in the intermediate range.

Abstract

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.