Turán problems and shadows II: Trees

Abstract 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 ex 3 ( n , G + ) when G is a path or cycle, thus settling conjectures of Furedi–Jiang [8] (for cycles) and Furedi–Jiang–Seiver [9] (for paths). Here we continue this project by determining the asymptotics for ex 3 ( n , G + ) when G is any fixed forest. This settles a conjecture of Furedi [7] . Using our methods, we also show that for any graph G, either ex 3 ( n , G + ) ≤ ( 1 2 + o ( 1 ) ) n 2 or ex 3 ( n , G + ) ≥ ( 1 + o ( 1 ) ) n 2 , thereby exhibiting a jump for the Turan number of expansions.