A survey of Tur\'an problems for expansions

TL;DR

This survey reviews recent advances in Turán problems for expansions of graphs, focusing on extremal hypergraph problems where the goal is to determine maximum edge counts avoiding certain expanded subgraphs.

Contribution

It compiles and discusses recent progress in understanding Turán numbers for graph expansions in hypergraphs, highlighting new results and open problems.

Findings

Summarizes key results on Turán numbers for expansions

Identifies open problems in extremal hypergraph theory

Highlights recent techniques and approaches

Abstract

The $r$-expansion $G^+$ of a graph $G$ is the $r$-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a vertex subset of size $r-2$ disjoint from $V(G)$ such that distinct edges are enlarged by disjoint subsets. Let $ex_r(n,F)$ denote the maximum number of edges in an $r$-uniform hypergraph with $n$ vertices not containing any copy of the $r$-uniform hypergraph $F$. Many problems in extremal set theory ask for the determination of $ex_r(n,G^+)$ for various graphs $G$. We survey these Tur\'an-type problems, focusing on recent developments.