Let $N_{\mathcal{P}}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. The problem of bounding this function for various graphs $H$ has been extensively studied since the 70's. A special case that received a lot of attention recently is when $H$ is the path on $2m+1$ vertices, denoted $P_{2m+1}$. Our main result is that $$ N_{\mathcal{P}}(n,P_{2m+1})=O(m^{-m}n^{m+1})\;. $$ This improves upon the previously best known bound by a factor $e^{m}$, which is best possible up to the hidden constant, and makes a significant step towards resolving conjectures of Gosh et al. and of Cox and Martin. The proof uses graph theoretic arguments together with (simple) arguments from the theory of convex optimization.
Joint work with Asaf Shapira.