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  recently determined ex3(n,G+) when G is a path or cycle, thus settling conjectures of Füredi–Jiang  (for cycles) and Füredi–Jiang–Seiver  (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 . 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.
- Expansions of graphs
- Hypergraph Turán numbers
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics