Survey of Generalized Turán Problems — Counting Subgraphs

Abstract

For fixed graphs $H$ and $F$, the generalized Turán number $\mathrm{ex}(n,H,F)$ is the maximum possible number of copies of a subgraph $H$ in an $n$-vertex $F$-free graph. This article is a survey of this extremal function whose study was initiated in an influential 2016 article by Alon and Shikhelm [J. Combin. Theory, B, 121, 2016].