A survey of Turán problems for expansions

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 $e{x}_r(n,F)$ 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 $e{x}_r(n,G^{+})$ for various graphs $G$. We survey these Turán-type problems, focusing on recent developments.