Regular Partitions of Hypergraphs: Regularity Lemmas

Szemeredi's regularity lemma for graphs has proved to be a powerful tool with many subsequent applications. The objective of this paper is to extend the techniques developed by Nagle, Skokan, and the authors and obtain a stronger and more ‘user-friendly’ regularity lemma for hypergraphs.

[1]  H. Furstenberg,et al.  An ergodic Szemerédi theorem for IP-systems and combinatorial theory , 1985 .

[2]  Noga Alon,et al.  Efficient Testing of Large Graphs , 2000, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[3]  Vojtech Rödl,et al.  The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent , 1986, Graphs Comb..

[4]  Vojtech Rödl,et al.  Every Monotone 3-Graph Property is Testable , 2007, SIAM J. Discret. Math..

[5]  Yoshiharu Kohayakawa,et al.  Hypergraphs, Quasi-randomness, and Conditions for Regularity , 2002, J. Comb. Theory, Ser. A.

[6]  Vojtech Rödl,et al.  On the Ramsey Number of Sparse 3-Graphs , 2008, Graphs Comb..

[7]  B. Green A Szemerédi-type regularity lemma in abelian groups, with applications , 2003, math/0310476.

[8]  Vojtech Rödl,et al.  A sharp threshold for random graphs with a monochromatic triangle in every edge coloring , 2006, Memoirs of the American Mathematical Society.

[9]  Daniela Kühn,et al.  Embeddings and Ramsey numbers of sparse κ-uniform hypergraphs , 2006, Comb..

[10]  Vojtech Rödl,et al.  Regularity Lemma for k‐uniform hypergraphs , 2004, Random Struct. Algorithms.

[11]  Vojtech Rödl,et al.  Integer and fractional packings of hypergraphs , 2007, J. Comb. Theory, Ser. B.

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

[13]  N. Alon,et al.  testing of large graphs , 2000 .

[14]  W. T. Gowers,et al.  Lower bounds of tower type for Szemerédi's uniformity lemma , 1997 .

[15]  József Solymosi,et al.  A Note on a Question of Erdős and Graham , 2004, Combinatorics, Probability and Computing.

[16]  Vojtech Rödl,et al.  Applications of the regularity lemma for uniform hypergraphs , 2006, Random Struct. Algorithms.

[17]  H. Furstenberg,et al.  An ergodic Szemerédi theorem for commuting transformations , 1978 .

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

[19]  Vojtech Rödl,et al.  The counting lemma for regular k‐uniform hypergraphs , 2006, Random Struct. Algorithms.

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

[21]  E. Szemerédi On sets of integers containing k elements in arithmetic progression , 1975 .

[22]  János Komlós,et al.  The Regularity Lemma and Its Applications in Graph Theory , 2000, Theoretical Aspects of Computer Science.

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

[24]  RodlVojtech,et al.  Regular Partitions of Hypergraphs , 2007 .

[25]  Mathias Schacht,et al.  Density theorems and extremal hypergraph problems , 2006 .

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

[27]  Vojtech Rödl,et al.  Extremal problems on set systems , 2002, Random Struct. Algorithms.

[28]  Jozef Skokan,et al.  Applications of the regularity lemma for uniform hypergraphs , 2006 .

[29]  Vojtech Rödl,et al.  Every Monotone 3-Graph Property is Testable , 2005, SIAM J. Discret. Math..

[30]  V. Rödl,et al.  Extremal Hypergraph Problems and the Regularity Method , 2006 .

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

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

[33]  Y. Kohayakawa Szemerédi's regularity lemma for sparse graphs , 1997 .

[34]  H. Furstenberg,et al.  A density version of the Hales-Jewett theorem , 1991 .