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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 variousgraphs $H$ has been extensively studied since the 70's. A special case thatreceived 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 thepreviously best known bound by a factor $e^{m}$, which is best possible up tothe hidden constant, and makes a significant step towards resolving conjecturesof Gosh et al. and of Cox and Martin. The proof uses graph theoretic argumentstogether with (simple) arguments from the theory of convex optimization.

Bounding the number of odd paths in planar graphs via convex optimization