The Generalized Rainbow Turán Problem for Cycles

Given an edge-coloured graph, we say that a subgraph is rainbow if all of its edges have different colours. Let $\operatorname{ex}(n,H,$rainbow-$F)$ denote the maximal number copies $H$ properly graph on $n$ vertices can contain it has no isomorphic to $F$. We determine order magnitude $\operatorname{ex}(n,C_s,$rainbow-$C_t)$ for $s,t$ with $s\not =3$. In particular, answer question Gerbner, M\'esz\'aros, Methuku and Palmer by showing $\operatorname{ex}(n,C_{2k},$rainbow-$C_{2k})$ $\Theta(n^{k-1})$ $k\geq 3$ $\Theta(n^2)$ $k=2$. also $\operatorname{ex}(n,P_\ell,$rainbow-$C_{2k})$ $k,\ell\geq 2$, where $P_\ell$ denotes path $\ell$ edges.