The Multicolour Ramsey Number of a Long Odd Cycle

For a graph G, the k  -colour Ramsey number Rk(G)Rk(G) is the least integer N such that every k  -colouring of the edges of the complete graph KNKN contains a monochromatic copy of G  . Bondy and Erdős conjectured that for an odd cycle CnCn on n>3n>3 vertices,Rk(Cn)=2k−1(n−1)+1.Rk(Cn)=2k−1(n−1)+1. This is known to hold when k=2k=2 and n>3n>3, and when k=3k=3 and n   is large. We show that this conjecture holds asymptotically for k≥4k≥4, proving that for n odd,Rk(Cn)=2k−1n+o(n)asn→∞. The proof uses the regularity lemma to relate this problem in Ramsey theory to one in convex optimisation, allowing analytic methods to be exploited. Our analysis leads us to a new class of lower bound constructions for this problem, which naturally arise from perfect matchings in the k-dimensional hypercube. Progress towards a resolution of the conjecture for large n is also discussed.