  Given graphs $G, H$ and an integer $q \ge 2$, the generalized Ramsey number,
denoted $r(G,H,q)$, is the minimum number of colours needed to edge-colour $G$
such that every copy of $H$ receives at least $q$ colours. In this paper, we
prove that for a fixed integer $k \ge 3$, we have $r(K_n,C_k,3) =
n/(k-2)+o(n)$. This generalises work of Joos and Muybayi, who proved
$r(K_n,C_4,3) = n/2+o(n)$. We also provide an upper bound on $r(K_{n,n}, C_k,
3)$, which generalises a result of Joos and Mubayi that $r(K_{n,n},C_4,3) =
2n/3+o(n)$. Both of our results are in fact specific cases of more general
theorems concerning families of cycles.
