Applications of the regularity lemma for uniform hypergraphs

In this article we discuss several combinatorial problems that can be addressed by the Regularity Method for hypergraphs. Based on the recent results of Nagle, Schacht, and the authors, we give here solutions to these problems.In particular, we prove the following: Let F be a k-uniform hypergraph on t vertices and suppose an n-vertex k-uniform hypergraph H contains only o(nt) copies of F. Then one can delete o(nk) edges of H to make it F-free.Similar results were recently obtained by W. T. Gowers. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

[1]  I. Bárány.  LECTURES ON DISCRETE GEOMETRY (Graduate Texts in Mathematics 212) , 2003 .

[2]  P. Erdös.  On extremal problems of graphs and generalized graphs , 1964 .

[3]  Vojtech Rödl,et al.  Regularity properties for triple systems , 2003, Random Struct. Algorithms.

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

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

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

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

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

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

[10]  I. Shparlinski.  On a question of Erdös and Graham , 2002 .

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

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

[13]  Mathias Schacht,et al.  On the Regularity Method for Hypergraphs , 2004 .

[14]  V. Rödl,et al.  The counting lemma for regular k-uniform hypergraphs , 2006 .

[15]  Z. Füredi.  Extremal Hypergraphs and Combinatorial Geometry , 1995 .

[16]  V. Rödl,et al.  Extremal problems on set systems , 2002 .

[17]  Paul Erdös,et al.  On Some Sequences of Integers , 1936 .

[18]  Vojtech Rödl,et al.  Counting subgraphs in quasi-random 4-uniform hypergraphs , 2005, Random Struct. Algorithms.

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

[20]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[21]  Jozef Skokan,et al.  Counting subgraphs in quasi-random 4-uniform hypergraphs , 2005 .

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

[23]  P. Erdös,et al.  On the structure of linear graphs , 1946 .

[24]  W. G. Brown,et al.  On the existence of triangulated spheres in 3-graphs, and related problems , 1973 .

[25]  Miklós Simonovits,et al.  Supersaturated graphs and hypergraphs , 1983, Comb..

[26]  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..

[27]  Paul Erdös,et al.  On Pisier Type Problems and Results (Combinatorial Applications to Number Theory) , 1990 .