Generalized Tur\’an Problems for Trees and More

Given a graph $H$ and a family of graphs $\mathcal{F}$, we define the generalized Tur\’an number $\mathrm{ex}(n,H,\mathcal{F})$ to be the maximum number of copies of $H$ in an $\mathcal{F}$-free graph on $n$ vertices. We prove a “stability” type result for generalized Tur\’an problems which relates the generalized Tur\’an number $\mathrm{ex}(n,H,\mathcal{F})$ to the classical Tur\’an number $\mathrm{ex}(n,\mathcal{F})$ whenever $H$ is a tree. We discuss some applications of this result, as well as some related work around the rational exponents conjecture for general graphs $H$. Joint work with Sean English.

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