A hypergraph regularity method for generalized Turán problems

We describe a method that we believe may be foundational for a comprehensive theory of generalised Turán problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed. We demonstrate the power of the method by proving a conjecture of Mubayi on the codegree threshold of the Fano plane, that any 3-graph on n vertices for which every pair of vertices is contained in more than n/2 edges must contain a Fano plane, for n sufficiently large. For projective planes over fields of odd size q we show that the codegree threshold is between n/2 − q + 1 and n/2, but for PG2(4) we find the somewhat surprising phenomenon that the threshold is less than (1/2 − ǫ)n for some small ǫ > 0. We conclude by setting out a program for future developments of this method to tackle other problems.

[1]  Fan Chung Graham,et al.  Quasi-random graphs , 1988, Comb..

[2]  T. Tao.  Arithmetic progressions and the primes - El Escorial lectures , 2004, math/0411246.

[3]  L. Berardi,et al.  Blocking sets in the projective plane of order four , 1986 .

[4]  Yi Zhao,et al.  Codegree problems for projective geometries , 2007, J. Comb. Theory, Ser. B.

[5]  E. Szemerédi.  Regular Partitions of Graphs , 1975 .

[6]  G. Tallini.  On Blocking Sets in Finite Projective and Affine Spaces , 1988 .

[7]  Daniela Kühn,et al.  Loose Hamilton cycles in 3-uniform hypergraphs of high minimum degree , 2006, J. Comb. Theory, Ser. B.

[8]  Vojtech Rödl,et al.  Regular Partitions of Hypergraphs: Regularity Lemmas , 2007, Combinatorics, Probability and Computing.

[9]  M. Simonovits,et al.  Szemeredi''s Regularity Lemma and its applications in graph theory , 1995 .

[10]  Terence Tao.  A variant of the hypergraph removal lemma , 2006, J. Comb. Theory, Ser. A.

[11]  Fan Chung Graham,et al.  Quasi-Random Hypergraphs , 1990, Random Struct. Algorithms.

[12]  Dhruv Mubayi,et al.  Set systems without a simplex or a cluster , 2010, Comb..

[13]  Zoltán Füredi,et al.  The Maximum Size of 3-Uniform Hypergraphs Not Containing a Fano Plane , 2000, J. Comb. Theory, Ser. B.

[14]  László Lovász,et al.  Limits of dense graph sequences , 2004, J. Comb. Theory B.

[15]  Vojtech Rödl,et al.  A Dirac-Type Theorem for 3-Uniform Hypergraphs , 2006, Combinatorics, Probability and Computing.

[16]  Dhruv Mubayi.  The co-degree density of the Fano plane , 2005, J. Comb. Theory, Ser. B.

[17]  W. T. Gowers,et al.  Hypergraph regularity and the multidimensional Szemerédi theorem , 2007, 0710.3032.

[18]  T. Tao,et al.  The primes contain arbitrarily long arithmetic progressions , 2004, math/0404188.

[19]  Miklós Simonovits,et al.  Triple Systems Not Containing a Fano Configuration , 2005, Comb. Probab. Comput..

[20]  W. T. Gowers,et al.  Quasirandomness, Counting and Regularity for 3-Uniform Hypergraphs , 2006, Combinatorics, Probability and Computing.

[21]  Vojtech Rödl,et al.  Property testing in hypergraphs and the removal lemma , 2007, STOC '07.

[22]  Peter Keevash.  The Turán problem for projective geometries , 2005, J. Comb. Theory, Ser. A.

[23]  J. Hirschfeld.  Projective Geometries Over Finite Fields , 1980 .

[24]  Benny Sudakov,et al.  On A Hypergraph Turán Problem Of Frankl , 2005, Comb..

[25]  Alexander A. Razborov,et al.  Flag algebras , 2007, Journal of Symbolic Logic.

[26]  Yi Zhao,et al.  Co-degree density of hypergraphs , 2007, J. Comb. Theory, Ser. A.