Set-coloring Ramsey numbers and error-correcting codes near the zero-rate threshold
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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 s/r is less than and bounded away from 1-1/n-1, then R(n;r,s) is known to grow exponentially in r, while if s/r is greater than and bounded away from 1-1/n-1, then R(n;r,s) is bounded. Here we prove bounds for R(n;r,s) in the intermediate range where s/r is close to 1 - 1/n-1 by establishing a connection to the maximum size of error-correcting codes near the zero-rate threshold.
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