Multicolour Ramsey numbers of paths and even cycles

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)⩽Rk(Pn)⩽Rk(Cn)⩽kn+o(n). The upper bound was recently improved by Sárközy who showed that Rk(Cn)⩽k−k16k3+1n+o(n). Here we show Rk(Cn)⩽(k−14)n+o(n), obtaining the first improvement to the coefficient of the linear term by an absolute constant.