  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.
