The Multicolour Ramsey Number of a Long Odd Cycle

Abstract For a graph G, the k-colour Ramsey number R k ( G ) is the least integer N such that every k-colouring of the edges of the complete graph K N contains a monochromatic copy of G. Bondy and Erdős conjectured that for an odd cycle C n on n > 3 vertices, R k ( C n ) = 2 k − 1 ( n − 1 ) + 1 . This is known to hold when k = 2 and n > 3 , and when k = 3 and n is large. We show that this conjecture holds asymptotically for k ≥ 4 , proving that for n odd, R k ( C n ) = 2 k − 1 n + o ( n ) as n → ∞ . 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.