Maximum packings in graphs forbidding given rainbow cycles

Motivated by the Ruzsa-Szemer\'{e}di problem, Imolay, Karl, Nagy, and V\'{a}li studied a variant of Tur\'{a}n number $ex_F(n,G)$ (called the $F$-multicolor Tur\'{a}n number of $G$), defined as the maximum number of edge-disjoint copies of $F$ on $n$-vertex set such that there is no copies of $G$ whose edges come from distinct copies of $F$. They proved that if there is no homomorphism from $G$ to $F$, then $n^2/v(F)^2+o(n^2)\leq ex_F(n,G)\leq ex(n,G)/e(F)+o(n^2)$, and otherwise $ex_F(n,G) = o(n^2)$. The quantity $ex_F(n,G)$ asymptotically equals the maximum size of an $F$-packing in an $n$-vertex $G$-free graph, and attains the upper bound $ex(n,G)/e(F)+o(n^2)$ if and only if $\chi(G) > \chi(F)$. In this paper, we provide conditions under which $ex_F(n,G)$ does not achieve the lower bound $n^2/v(F)^2 + o(n^2)$, and describe additional graph pairs that attain this lower bound via graph blow-ups. Especially, we proved that $ex_{C_k(s)}(n,C_{k-2})=n^2/(sk)^2+o(n^2)$ for any $k\geq 5$. For degenerate cases, we show that if $\chi(F) = 3$ and $G$ and $F$ share the same odd girth, then $ex_F(n,G)$ satisfies the $(6,3)$-type bound $n^{2-o(1)}$, generalizing a result of Kov\'acs and Nagy. We also prove that $ex_{C_{2k+1}}(n,C_{2\ell+1})=O(n^{1+1/(\ell-k+1)})$ for any integers $k,\ell$ with $\ell>k$, extending a result of F\"{u}redi and \"{O}zkahya. Additionally, we establish $ex_{C_4}(n,C_4)=\sqrt{2}n^{3/2}/8+O(n)$.