The Erd\H{o}s-Gy\'arf\'as problem on generalized Ramsey numbers

Graphs, Hypergraphs, and Computing

20 March 15:30 - 16:30

David Conlon - University of Oxford

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 Erd\H{o}s and Shelah about 40 years ago, but Erd\H{o}s and Gy\'arf\'as 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 $p-1$.

Joint work with Jacob Fox, Choongbum Lee and Benny Sudakov.
Magnus M. Halldorsson
Reykjavik University
Klas Markström
Umeå University
Andrzej Rucinski
Adam Mickiewicz University
Carsten Thomassen
Technical University of Denmark, DTU


Klas Markström


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