Abstract
Abstract The q-colour Ramsey number of a k-uniform hypergraph H is the minimum integer N such that any q-colouring of the complete k-uniform hypergraph on N vertices contains a monochromatic copy of H. The study of these numbers is one of the central topics in Combinatorics. In 1973, Erdős and Graham asked to maximise the Ramsey number of a graph as a function of the number of its edges. Motivated by this problem, we study the analogous question for hypergaphs. For fixed $k \ge 3$ and $q \ge 2$ we prove that the largest possible q-colour Ramsey number of a k-uniform hypergraph with m edges is at most $\mathrm{tw}_k(O(\sqrt{m})),$ where tw denotes the tower function. We also present a construction showing that this bound is tight for $q \ge 4$ . This resolves a problem by Conlon, Fox and Sudakov. They previously proved the upper bound for $k \geq 4$ and the lower bound for $k=3$ . Although in the graph case the tightness follows simply by considering a clique of appropriate size, for higher uniformities the construction is rather involved and is obtained by using paths in expander graphs.
Submitted Version (
Free)
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call