Survey of generalized Tur\'an problems -- counting subgraphs

TL;DR

This survey reviews the generalized Turán problem, which involves determining the maximum number of subgraphs of a fixed type in large graphs that avoid another fixed subgraph, highlighting developments since 2016.

Contribution

It compiles and discusses recent progress and open problems in the study of generalized Turán numbers for counting subgraphs in extremal graph theory.

Findings

Summarizes key results and conjectures in generalized Turán problems.

Identifies gaps and directions for future research.

Provides a comprehensive overview of the field since 2016.

Abstract

For fixed graphs $H$ and $F$, the \emph{generalized Tur\'an 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 Shikhelman (\emph{J. Combin. Theory, B}, {\bf 121}, 2016).