Turán Problems and Shadows III: Expansions of Graphs

The expansion G + of a graph G is the 3-uniform hypergraph obtained from G by enlarging each edge of G with a new vertex disjoint from V (G) such that distinct edges are enlarged by distinct vertices. Let ex3(n, F ) denote the maximum number of edges in a 3-uniform hypergraph with n vertices not containing any copy of a 3-uniform hypergraph F. The study of ex3(n, G + ) includes some well-researched problems, including the case that F consists of k disjoint edges, G is a triangle, G is a path or cycle, and G is a tree. In this paper we initiate a broader study of the behavior of ex3(n, G+). Specifically, we show ex3(n, K + )=Θ (n 3−3/s) whenever t> (s − 1)! and s ≥ 3. One of the main open problems is to determine for which graphs G the quantity ex3(n, G + ) is quadratic in n. We show that this occurs when G is any bipartite graph with Turan number o(n ϕ ) where ϕ = 1+ √ 5 2 , and in particular this shows ex3(n, G + )= O(n 2 )w henG is the three-dimensional cube graph.

[1]  Béla Bollobás,et al.  Extremal problems in graph theory , 1977, J. Graph Theory.

[2]  Rita Csákány,et al.  A Homological Approach to Two Problems on Finite Sets , 1999 .

[3]  Alexandr V. Kostochka,et al.  Turán problems and shadows II: Trees , 2014, J. Comb. Theory, Ser. B.

[4]  Zoltán Füredi,et al.  Linear trees in uniform hypergraphs , 2012, Eur. J. Comb..

[5]  Noga Alon,et al.  Turán Numbers of Bipartite Graphs and Related Ramsey-Type Questions , 2003, Combinatorics, Probability and Computing.

[6]  Zoltán Füredi,et al.  Exact solution of the hypergraph Turán problem for k-uniform linear paths , 2011, Comb..

[7]  Zoltán Füredi,et al.  Hypergraph Turán numbers of linear cycles , 2014, J. Comb. Theory, Ser. A.

[8]  Jacques Verstraëte,et al.  A hypergraph extension of the bipartite Turán problem , 2004, J. Comb. Theory, Ser. A.

[9]  Christopher M. Hartman.  Extremal problems in graph theory , 1997 .

[10]  Gyula O. H. Katona,et al.  Intersection theorems for systems of finite sets , 1964 .

[11]  Oleg Pikhurko.  Exact computation of the hypergraph Turán function for expanded complete 2-graphs , 2013, J. Comb. Theory, Ser. B.

[12]  V. Sós,et al.  On a problem of K. Zarankiewicz , 1954 .

[13]  Alexandr V. Kostochka,et al.  Turán problems and shadows I: Paths and cycles , 2013, J. Comb. Theory, Ser. A.

[14]  Jacques Verstraëte,et al.  Proof Of A Conjecture Of Erdős On Triangles In Set-Systems , 2005, Comb..

[15]  Dhruv Mubayi,et al.  A hypergraph extension of Turán's theorem , 2006, J. Comb. Theory, Ser. B.

[16]  W. G. Brown.  On Graphs that do not Contain a Thomsen Graph , 1966, Canadian Mathematical Bulletin.

[17]  Zoltán Füredi,et al.  Exact solution of some Turán-type problems , 1987, J. Comb. Theory, Ser. A.

[18]  Noga Alon,et al.  Norm-Graphs: Variations and Applications , 1999, J. Comb. Theory, Ser. B.

[19]  Peter Frankl.  On families of finite sets no two of which intersect in a singleton , 1977, Bulletin of the Australian Mathematical Society.

[20]  N. Alon,et al.  Triangles in H-free graphs , 2014 .