Density Hales-Jewett and Moser Numbers

For any n ≥ 0 and k ≤ 1, the density Hales-Jewett number cn,k is defined as the size of the largest subset of the cube [k]n:= 1,..., kn which contains no combinatorial line; similarly, the Moser number c nk is the largest subset of the cube [k]n which contains no geometric line. A deep theorem of Furstenberg and Katznelson [11], [12], [19] shows that c n,k =o(k n ) as n→∞ (which implies a similar claim for cn); this is already non-trivial for k = 3. Several new proofs of this result have also been recently established [23], [2].

[1]  K. F. Roth.  On Certain Sets of Integers , 1953 .

[2]  Tim Austin,et al.  Deducing the Density Hales–Jewett Theorem from an Infinitary Removal Lemma , 2009, 0903.1633.

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

[4]  S. R. Simanca,et al.  On Circulant Matrices , 2012 .

[5]  M. Narasimha Murty,et al.  Genetic K-means algorithm , 1999, IEEE Trans. Syst. Man Cybern. Part B.

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

[7]  Michael Elkin.  An improved construction of progression-free sets , 2010, SODA '10.

[8]  Aaron Potechin.  Maximal caps in AG (6, 3) , 2008, Des. Codes Cryptogr..

[9]  J. Beck.  Combinatorial Games: Tic-Tac-Toe Theory , 2008 .

[10]  A. K. Chandra.  On the Solution of Moser's Problem in Four Dimensions , 1973, Canadian Mathematical Bulletin.

[11]  R. A. Rankin,et al.  XXIV.—Sets of Integers Containing not more than a Given Number of Terms in Arithmetical Progression , 1961, Proceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences.

[12]  V. Chvatal,et al.  Remarks on a Problem of Moser , 1972, Canadian Mathematical Bulletin.

[13]  F. Behrend.  On Sets of Integers Which Contain No Three Terms in Arithmetical Progression. , 1946, Proceedings of the National Academy of Sciences of the United States of America.

[14]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[15]  Michael Elkin.  An improved construction of progression-free sets , 2010, SODA 2010.

[16]  Saharon Shelah,et al.  Primitive recursive bounds for van der Waerden numbers , 1988 .

[17]  E. Sperner.  Ein Satz über Untermengen einer endlichen Menge , 1928 .

[18]  Vasek Chvátal,et al.  Edmonds polytopes and a hierarchy of combinatorial problems , 1973, Discret. Math..

[19]  Julia Wolf,et al.  A Note on Elkin’s Improvement of Behrend’s Construction , 2010 .

[20]  D. Polymath,et al.  A new proof of the density Hales-Jewett theorem , 2009, 0910.3926.

[21]  Y. Katznelson,et al.  A density version of the Hales-Jewett theorem for k=3 , 1989, Discret. Math..

[22]  Knut Bahr,et al.  Solution to problem 8 , 1975, SIGS.

[23]  Franz Rothlauf,et al.  Representations for genetic and evolutionary algorithms , 2002, Studies in Fuzziness and Soft Computing.

[24]  Kevin O'Bryant,et al.  Sets of Integers that do not Contain Long Arithmetic Progressions , 2008, Electron. J. Comb..