Turán problems and shadows II: Trees

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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 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 [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+)(12+o(1))n2 or ex3(n,G+)(1+o(1))n2, thereby exhibiting a jump for the Turán number of expansions.

Keywords

Hypergraph Turán numbers
Expansions of graphs
Forests
Crosscuts

Cited by (0)

1

Research of this author is supported in part by NSF grants DMS-1266016 and DMS-1600592 and by Grant NSh.1939.2014.1 of the President of Russia for Leading Scientific Schools.

2

Research partially supported by NSF grants DMS-0969092 and DMS-1300138.

3

Research supported by NSF Grant DMS-1101489.