Sam Spiro, Georgia State University

Abstract

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.