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arXiv · 2305.14132

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

David Conlon·Jacob Fox·Huy Tuan Pham·Yufei Zhao
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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.

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David Conlon, Jacob Fox, Huy Tuan Pham, Yufei Zhao. 2023-05-23. Set-coloring Ramsey numbers and error-correcting codes near the zero-rate threshold. https://arxiv.org/abs/2305.14132

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