Structure and Randomness in Combinatorics

Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has proven profitable to decompose such objects into a structured component, a pseudo-random component, and a small component (i.e. an error term): in many cases it is the structured component which then dominates. We illustrate this philosophy in a number of model cases.

[1]  W. T. Gowers,et al.  A New Proof of Szemerédi's Theorem for Arithmetic Progressions of Length Four , 1998 .

[2]  Ben Green,et al.  On the Littlewood Problem Modulo a Prime , 2009, Canadian Journal of Mathematics.

[3]  A SZEMERI DI TYPE THEOREM FOR SETS OF POSITIVE DENSITY IN R k , 2022 .

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

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

[6]  Terence Tao,et al.  Additive combinatorics , 2007, Cambridge studies in advanced mathematics.

[7]  T. Tao,et al.  The primes contain arbitrarily long polynomial progressions , 2006, math/0610050.

[8]  Ben Green,et al.  AN INVERSE THEOREM FOR THE GOWERS $U^3(G)$ NORM , 2008, Proceedings of the Edinburgh Mathematical Society.

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

[11]  Terence Tao,et al.  A Correspondence Principle between (hyper)graph Theory and Probability Theory, and the (hyper)graph Removal Lemma , 2006 .

[12]  Alan M. Frieze,et al.  Quick Approximation to Matrices and Applications , 1999, Comb..

[13]  Ben Green,et al.  Montreal Lecture Notes on Quadratic Fourier Analysis , 2006 .

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

[15]  Terence Tao,et al.  The Gaussian primes contain arbitrarily shaped constellations , 2005 .

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

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

[18]  Terence Tao.  Szemerédi's regularity lemma revisited , 2006, Contributions Discret. Math..

[19]  W. T. Gowers,et al.  A new proof of Szemerédi's theorem , 2001 .

[20]  Manuel Blum,et al.  Self-testing/correcting with applications to numerical problems , 1990, STOC '90.

[21]  Noga Alon,et al.  Testing Low-Degree Polynomials over GF(2( , 2003, RANDOM-APPROX.

[22]  Terence Tao.  A Quantitative Ergodic Theory Proof of Szemerédi's Theorem , 2006, Electron. J. Comb..

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

[24]  Ben Green,et al.  New bounds for Szemerédi's theorem, I: progressions of length 4 in finite field geometries , 2009 .

[25]  T. Tao.  The ergodic and combinatorial approaches to Szemerédi's theorem , 2006, math/0604456.

[26]  Terence Tao,et al.  The dichotomy between structure and randomness, arithmetic progressions, and the primes , 2005, math/0512114.

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

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

[29]  Terence Tao,et al.  Norm convergence of multiple ergodic averages for commuting transformations , 2007, Ergodic Theory and Dynamical Systems.