The expansion of a graph G is the 3-uniform hypergraph obtained from G by enlarging each edge of G with a vertex disjoint from such that distinct edges are enlarged by distinct vertices. Let denote the maximum number of edges in an r-uniform hypergraph with n vertices not containing any copy of F. The authors [10] recently determined when G is a path or cycle, thus settling conjectures of Füredi–Jiang [8] (for cycles) and Füredi–Jiang–Seiver [9] (for paths).
Here we continue this project by determining the asymptotics for when G is any fixed forest. This settles a conjecture of Füredi [7]. Using our methods, we also show that for any graph G, either or , thereby exhibiting a jump for the Turán number of expansions.
Research of this author is supported in part by NSF grants DMS-1266016 and DMS-1600592 and by Grant NSh.1939.2014.1 of the President of Russia for Leading Scientific Schools.