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 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 Turan-type problems, focusing on recent developments.