Multicolour Ramsey numbers of paths and even cycles

TL;DR

This paper improves the upper bounds on multicolour Ramsey numbers for paths and even cycles, notably reducing the coefficient of the linear term in the bounds for cycles.

Contribution

It presents a new upper bound for the multicolour Ramsey numbers of even cycles, improving the linear coefficient by an absolute constant.

Findings

New upper bound: R_k(C_n) ≤ (k - 1/4)n + o(n)

First improvement to the linear coefficient by an absolute constant

Advances understanding of multicolour Ramsey numbers for cycles

Abstract

We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that $(k-1)n+o(n)\leq R_k(P_n)\leq R_k(C_n)\leq kn+o(n)$. The upper bound was recently improved by S\'ark\"ozy who showed that $R_k(C_n)\leq\left(k-\frac{k}{16k^3+1}\right)n+o(n)$. Here we show $R_k(C_n) \leq (k-\frac14)n +o(n)$, obtaining the first improvement to the coefficient of the linear term by an absolute constant.