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

Let (Formula presented.) denote the maximum number of copies of (Formula presented.) in an (Formula presented.) vertex planar graph. The problem of bounding this function for various graphs (Formula presented.) has been extensively studied since the 70's. A special case that received a lot of attention recently is when (Formula presented.) is the path on (Formula presented.) vertices, denoted (Formula presented.). Our main result in this paper is that (Formula presented.). This improves upon the previously best known bound by a factor (Formula presented.), which is best possible up to the hidden constant, and makes a significant step toward resolving conjectures of Ghosh et al. and of Cox and Martin. The proof uses graph theoretic arguments together with (simple) arguments from the theory of convex optimization.