Generalized Tur\'an problems for even cycles

Given a graph $H$ and a set of graphs $\mathcal{F}$, let $ex(n,H,\mathcal{F})$ denote the maximum possible number of copies of $H$ in an $\mathcal{F}$-free graph on $n$ vertices. We investigate the function $ex(n,H,\mathcal{F})$, when $H$ and members of $\mathcal{F}$ are cycles. Let $C_k$ denote the cycle of length $k$ and let ${\mathcal{C}}_k=\{C_3,C_4,\dots ,C_k\}$. Some of our main results are the following. (i) We show that $ex(n,C_{2l},C_{2k})=\mathrm{\Theta }(n^l)$ for any $l,k\ge 2$. Moreover, we determine it asymptotically in the following cases: We show that $ex(n,C_4,C_{2k})=(1+o(1))\frac{(k-1)(k-2)}{4}{n}^2$ and that the maximum possible number of $C_6$’s in a $C_8$-free bipartite graph is $n^3 + O(n^{5/2})$. (ii) Solymosi and Wong proved that if Erd\H{o}s’s Girth Conjecture holds, then for any $l\ge 3$ we have $ex(n,C_{2l},\mathscr C_{2l-1})=\Theta(n^{2l/(l-1)})$. We prove that forbidding any other even cycle decreases the number of $C_{2l}$'s significantly: For any $k>l$, we have $ex(n,C_{2l},{\mathcal{C}}_{2l-1}\cup \{C_{2k}\})=\mathrm{\Theta }(n^2)\mathrm{.}$ More generally, we show that for any $k>l$ and $m\ge 2$ such that $2k\mathrm{\ne }ml$, we have $ex(n,C_{ml},{\mathcal{C}}_{2l-1}\cup \{C_{2k}\})=\mathrm{\Theta }(n^m)\mathrm{.}$ (iii) We prove $ex(n,C_{2l+1},{\mathcal{C}}_{2l})=\mathrm{\Theta }(n^{2+1\mathrm{/}l}),$ provided a strong version of Erd\H{o}s’s Girth Conjecture holds (which is known to be true when $l=2,3,5$). Moreover, forbidding one more cycle decreases the number of $C_{2l+1}$'s significantly: More precisely, we have $ex(n, C_{2l+1}, \mathscr C_{2l} \cup {C_{2k}}) = O(n^{2-\frac{1}{l+1}}),$and$ex(n, C_{2l+1}, \mathscr C_{2l} \cup {C_{2k+1}}) = O(n^2)$for$l > k \ge 2$. (iv) We also study the maximum number of paths of given length in a $C_k$-free graph, and prove asymptotically sharp bounds in some cases.