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For graphs $H$ and $F$, let $\text{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Gerbner, Gy\H{o}ri, Methuku, and Vizer proved that $\text{ex}(n,C_{2\ell},C_{2k})=\Theta(n^\ell)$ for $k>\ell\ge2$. They determined the leading term for $\ell=2$, but for $k>\ell\ge3$ their general...
For graphs $G$ and $H$, let $\mathbf N(G,H)$ denote the number of unlabeled, not necessarily induced copies of $H$ in $G$, and let $\mathbf N_{\mathcal P}(n,H)$ be the maximum of $\mathbf N(G,H)$ over all $n$-vertex planar graphs $G$. We prove that, for every fixed integer $m\geq 3$, $$\mathbf N_{\mathcal P}(n,C_{2m+1}...
For a graph $H$ and a family of graphs $\mathcal F$, let $\text{ex}(n,H,\mathcal F)$ denote the maximum number of copies of $H$ in an $\mathcal F$-free graph on $n$ vertices. For every integer $i\ge 3$, let $C_i$ denote the cycle of length $i$. For $r\ge 3$, set $\mathscr {C}_r=\{C_3,C_4,\ldots,C_r\},$ and set $\mathsc...
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