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arXiv · 1403.0250

The Erdős-Gyárfás problem on generalized Ramsey numbers

David Conlon·Jacob Fox·Choongbum Lee·Benny Sudakov
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Abstract

Fix positive integers $p$ and $q$ with $2 \leq q \leq {p \choose 2}$. An edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ different colors. The function $f(n, p, q)$ is the minimum number of colors that are needed for $K_n$ to have a $(p,q)$-coloring. This function was introduced by Erdős and Shelah about 40 years ago, but Erdős and Gyárfás were the first to study the function in a systematic way. They proved that $f(n, p, p)$ is polynomial in $n$ and asked to determine the maximum $q$, depending on $p$, for which $f(n,p,q)$ is subpolynomial in $n$. We prove that the answer is $p-1$.

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David Conlon, Jacob Fox, Choongbum Lee, Benny Sudakov. 2014-03-02. The Erdős-Gyárfás problem on generalized Ramsey numbers. https://doi.org/10.1112/plms%2Fpdu049

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