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Algorithms, Combinatorics and Optimization Seminar
University of Oxford Title: On the Erdös–Gyárfás problem in generalised Ramsey theory Abstract: 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 Erdös and Shelah about 40 years ago, but Erdös 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. |