Exact Ramsey numbers of odd cycles via nonlinear optimisation

For a graph G, the k-colour Ramsey number Rk(G) is least integer N such that every k-colouring of edges complete KN contains monochromatic copy G. Let Cn denote cycle on n vertices. We show for fixed k?2 and odd sufficiently large,Rk(Cn)=2k?1(n?1)+1. This resolves conjecture Bondy Erd?s large n. The proof analytic in nature, first step which to use regularity method relate this problem theory one nonlinear optimisation. allows us prove stability-type generalisation above establish correspondence between extremal k-colourings perfect matchings k-dimensional hypercube Qk.