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 ex_r(n,F) 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 ex₃(n,G⁺) 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 ex₃(n,G⁺) 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 ex₃(n,G⁺)≤([formula presented]+o(1))n² or ex₃(n,G⁺)≥(1+o(1))n², thereby exhibiting a jump for the Turán number of expansions.