Abstract
AbstractLet denote the maximum number of copies of in an vertex planar graph. The problem of bounding this function for various graphs has been extensively studied since the 70's. A special case that received a lot of attention recently is when is the path on vertices, denoted . Our main result in this paper is that This improves upon the previously best known bound by a factor , 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.
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