A sharp asymptotic bound for odd cycles in planar graphs

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 $m\ge 3$, $${\mathbf{N}}_{\mathcal{P}}(n,C_{2m+1})=2m{\left(\frac{n}{m}\right)}^m+O_m\text{\hspace{0.17em}⁣}\left(n^{m-1\mathrm{/}5}\right)\mathrm{.}$$ The proof uses a sharp weighted cycle–path inequality for edge probability measures on finite complete graphs. This strengthens a conjecture of Heath, Martin, and Wells and, together with their reduction lemma, yields the stated asymptotic formula.