Exact Ramsey numbers of odd cycles via nonlinear optimisation

TL;DR

This paper determines the exact Ramsey numbers for odd cycles in multiple colours for large cycles, using nonlinear optimization and stability analysis, confirming a long-standing conjecture.

Contribution

It provides a proof of the exact Ramsey numbers for odd cycles in multiple colours for large n, employing nonlinear optimization and stability methods, and reveals a connection to hypercube matchings.

Findings

Exact formula for $R_k(C_n)$ for large odd n and fixed k

Establishment of a stability version of the main result

Identification of a correspondence between extremal colourings and hypercube matchings

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 $G$. Let $C_n$ denote the cycle on $n$ vertices. We show that for fixed $k\geq2$ and $n$ odd and sufficiently large, \[ R_k(C_n)=2^{k-1}(n-1)+1. \] This resolves a conjecture of Bondy and Erd\H{o}s [J. Combin. Th. Ser. B \textbf{14} (1973), 46--54] for large $n$. The proof is analytic in nature, the first step of which is to use the regularity method to relate this problem in Ramsey theory to one in nonlinear optimisation. This allows us to prove a stability-type generalisation of the above and establish a surprising correspondence between extremal $k$-colourings for this problem and perfect matchings in the $k$-dimensional hypercube $Q_k$.