  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 in this paper 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.

Combinatorics