Fix positive integers p and q with 2 ≤ q ≤ binom(p,2). An edge-colouring of the complete graph Kn is said to be a (p, q)-colouring if every Kp receives at least q different colours. The function f(n, p, q) is the minimum number of colours that are needed for Kn to have a (p,q)-colouring. 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.
We also discuss some related questions.
Joint work with Jacob Fox, Choongbum Lee and Benny Sudakov.
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