Counting paths, cycles and blow-ups in planar graphs

For a planar graph $H$, let $\operatorname{\mathbf{N}}_{\mathcal P}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. In this paper, we prove that $\operatorname{\mathbf{N}}_{\mathcal P}(n,P_7)\sim{4\over 27}n^4$, $\operatorname{\mathbf{N}}_{\mathcal P}(n,C_6)\sim(n/3)^3$, $\operatorname{\mathbf{N}}_{\mathcal P}(n,C_8)\sim(n/4)^4$ and $\operatorname{\mathbf{N}}_{\mathcal P}(n,K_4\{1\})\sim(n/6)^6$, where $K_4\{1\}$ is the $1$-subdivision of $K_4$. In addition, we obtain significantly improved upper bounds on $\operatorname{\mathbf{N}}_{\mathcal P}(n,P_{2m+1})$ and $\operatorname{\mathbf{N}}_{\mathcal P}(n,C_{2m})$ for $m\geq 4$. For a wide class of graphs $H$, the key technique developed in this paper allows us to bound $\operatorname{\mathbf{N}}_{\mathcal P}(n,H)$ in terms of an optimization problem over weighted graphs.