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arXiv · 2301.05678

A localized approach to generalized Tur\'an problems

Rachel Kirsch·JD Nir
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Abstract

Generalized Tur\'an problems ask for the maximum number of copies of a graph $H$ in an $n$-vertex, $F$-free graph, denoted by ex$(n,H,F)$. We show how to extend the new, localized approach of Brada\v{c}, Malec, and Tompkins to generalized Tur\'{a}n problems. We weight the copies of $H$ (typically taking $H=K_t$), instead of the edges, based on the size of the largest clique, path, or star containing the vertices of the copy of $H$, and in each case prove a tight upper bound on the sum of the weights. A consequence of our new localized theorems is an asymptotic determination of ex$(n,H,K_{1,r})$ for every $H$ having at least one dominating vertex and mex$(m,H,K_{1,r})$ for every $H$ having at least two dominating vertices.

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Rachel Kirsch, JD Nir. 2023-01-13. A localized approach to generalized Tur\'an problems. https://doi.org/10.37236/12132

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