Maximum packings in graphs forbidding given rainbow cycles

Motivated by the Ruzsa-Szemerédi problem, Imolay, Karl, Nagy, and Váli studied a variant of Turán number $e{x}_F(n,G)$ (called the $F$-multicolor Turán 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 $n^2\mathrm{/}v(F{)}^2+o(n^2)\le e{x}_F(n,G)\le ex(n,G)\mathrm{/}e(F)+o(n^2)$, and otherwise $e{x}_F(n,G)=o(n^2)$. The quantity $e{x}_F(n,G)$ asymptotically equals the maximum size of an $F$-packing in an $n$-vertex $G$-free graph, and attains the upper bound $ex(n,G)\mathrm{/}e(F)+o(n^2)$ if and only if $\chi (G)>\chi (F)$. In this paper, we provide conditions under which $e{x}_F(n,G)$ does not achieve the lower bound $n^2\mathrm{/}v(F{)}^2+o(n^2)$, and describe additional graph pairs that attain this lower bound via graph blow-ups. Especially, we proved that $e{x}_{C_k(s)}(n,C_{k-2})=n^2\mathrm{/}(sk{)}^2+o(n^2)$ for any $k\ge 5$ and its stability. For degenerate cases, we show that if $\chi (F)=3$ and $G$ and $F$ share the same odd girth, then $e{x}_F(n,G)$ satisfies the $(6,3)$-type bound $n^{2-o(1)}$, generalizing a result of Kovács and Nagy. We also prove that $e{x}_{C_{2k+1}}(n,C_{2\mathrm{\ell }+1})=O(n^{1+1\mathrm{/}\lceil \mathrm{\ell }\mathrm{/}k\rceil })$ for any integers $k,\mathrm{\ell }$ with $\mathrm{\ell }>k$, extending a result of Füredi and Özkahya, and also Collier-Cartaino, Graber, and Jiang. Additionally, we establish $e{x}_{C_4}(n,C_4)=\sqrt{2}{n}^{3\mathrm{/}2}\mathrm{/}8+O(n)$.