Turán problems and shadows II: Trees

The expansion G of a graph G is the 3-uniform hypergraph obtained from G by enlarging each edge of G with a vertex disjoint from V (G) such that distinct edges are enlarged by distinct vertices. Let exr(n, F ) denote the maximum number of edges in an r-uniform hypergraph with n vertices not containing any copy of F . The authors [11] recently determined ex3(n,G ) more generally, namely when G is a path or cycle, thus settling conjectures of Füredi-Jiang [9] (for cycles) and Füredi-Jiang-Seiver [10] (for paths). Here we continue this project by determining the asymptotics for ex3(n,G ) when G is any fixed forest. This settles a conjecture of Füredi [8]. Using our methods, we also show that for any graph G, either ex3(n,G ) ≤ ( 1 2 + o(1) ) n or ex3(n,G ) ≥ (1 + o(1))n, thereby exhibiting a jump for the Turán number of expansions.