## 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 Füredi–Jiang [8] (for cycles) and Füredi–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 Füredi [7]. Using our methods, we also show that for any graph G, either ex_{3}(n,G^{+})≤([formula presented]+o(1))n^{2} or ex_{3}(n,G^{+})≥(1+o(1))n^{2}, thereby exhibiting a jump for the Turán number of expansions.

Original language | English (US) |
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Pages (from-to) | 457-478 |

Number of pages | 22 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 122 |

DOIs | |

State | Published - Jan 1 2017 |

## Keywords

- Crosscuts
- Expansions of graphs
- Forests
- Hypergraph Turán numbers

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics