# Difference between revisions of "Polymath.tex"

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\author{D.H.J. Polymath} | \author{D.H.J. Polymath} | ||

\address{http://michaelnielsen.org/polymath1/index.php} | \address{http://michaelnielsen.org/polymath1/index.php} | ||

− | \email{ | + | %\email{} |

− | \subjclass{ | + | \subjclass{05D05, 05D10} |

\begin{abstract} | \begin{abstract} | ||

− | For any $n \geq 0$ and $k \geq 1$, the density Hales-Jewett number $c_{n,k}$ is defined as the size of the largest subset of the cube $[k]^n$ := $\{1,\ldots,k\}^n$ which contains no combinatorial line; similarly, the Moser number $c'_{n,k}$ is the largest subset of the cube $[k]^n$ which contains no geometric line. A deep theorem of Furstenberg and Katznelson \cite{fk1}, \cite{fk2}, \cite{mcc} shows that $c_{n,k}$ = $o(k^n)$ as $n \to \infty$ (which implies a similar claim for $c'_{n,k}$; this is already non-trivial for $k = 3$. Several new proofs of this result have also been recently established \cite{poly}, \cite{austin}. | + | For any $n \geq 0$ and $k \geq 1$, the \emph{density Hales-Jewett number} $c_{n,k}$ is defined as the size of the largest subset of the cube $[k]^n$ := $\{1,\ldots,k\}^n$ which contains no combinatorial line; similarly, the Moser number $c'_{n,k}$ is the largest subset of the cube $[k]^n$ which contains no geometric line. A deep theorem of Furstenberg and Katznelson \cite{fk1}, \cite{fk2}, \cite{mcc} shows that $c_{n,k}$ = $o(k^n)$ as $n \to \infty$ (which implies a similar claim for $c'_{n,k}$); this is already non-trivial for $k = 3$. Several new proofs of this result have also been recently established \cite{poly}, \cite{austin}. |

Using both human and computer-assisted arguments, we compute several values of $c_{n,k}$ and $c'_{n,k}$ for small $n,k$. For instance the sequence $c_{n,3}$ for $n=0,\ldots,6$ is $1,2,6,18,52,150,450$, while the sequence $c'_{n,3}$ for $n=0,\ldots,6$ is $1,2,6,16,43,124,353$. We also establish some results for higher $k$, showing for instance that an analogue of the LYM inequality (which relates to the $k = 2$ case) does not hold for higher $k$. | Using both human and computer-assisted arguments, we compute several values of $c_{n,k}$ and $c'_{n,k}$ for small $n,k$. For instance the sequence $c_{n,3}$ for $n=0,\ldots,6$ is $1,2,6,18,52,150,450$, while the sequence $c'_{n,3}$ for $n=0,\ldots,6$ is $1,2,6,16,43,124,353$. We also establish some results for higher $k$, showing for instance that an analogue of the LYM inequality (which relates to the $k = 2$ case) does not hold for higher $k$. | ||

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%\today | %\today | ||

− | \setcounter{tocdepth}{1} | + | %\setcounter{tocdepth}{1} |

− | \tableofcontents | + | %\tableofcontents |

− | + | ||

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% | ||

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\include{moser-lower} | \include{moser-lower} | ||

\include{moser} | \include{moser} | ||

− | \include{fujimura} | + | %\include{fujimura} |

− | \include{higherk} | + | %\include{higherk} |

− | \include{coloring} | + | % \include{coloring} |

− | \appendix | + | %\appendix |

− | \include{genetic} | + | %\include{genetic} |

− | \include{integer} | + | %\include{integer} |

\begin{thebibliography}{10} | \begin{thebibliography}{10} | ||

− | \bibitem{ajtai} | + | \bibitem{ajtai} M. Ajtai, E. Szemer\'edi, \emph{Sets of lattice points that form no squares}, Studia Scientiarum Mathematicarum Hungarica, 9, 9-11 (1974), (1975) |

\bibitem{austin} T. Austin, \emph{Deducing the density Hales-Jewett theorem from an infinitary removal lemma}, preprint. | \bibitem{austin} T. Austin, \emph{Deducing the density Hales-Jewett theorem from an infinitary removal lemma}, preprint. | ||

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\bibitem{behrend} | \bibitem{behrend} | ||

F. Behrend, \emph{On the sets of integers which contain no three in arithmetic progression}, Proceedings of the National Academy of Sciences \textbf{23} (1946), 331–-332. | F. Behrend, \emph{On the sets of integers which contain no three in arithmetic progression}, Proceedings of the National Academy of Sciences \textbf{23} (1946), 331–-332. | ||

+ | |||

+ | \bibitem{Brower} | ||

+ | A. Brower, {\tt www.win.tue.nl/$\sim$aeb/codes/binary-1.html}. | ||

\bibitem{chandra} | \bibitem{chandra} | ||

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\bibitem{elkin} | \bibitem{elkin} | ||

M. Elkin, \emph{An Improved Construction of Progression-Free Sets}, preprint. | M. Elkin, \emph{An Improved Construction of Progression-Free Sets}, preprint. | ||

+ | |||

+ | \bibitem{fuji} | ||

+ | K. Fujimura, {\tt www.puzzles.com/PuzzlePlayground/CoinsAndTriangles/CoinsAndTriangles.htm} | ||

\bibitem{fk1} H. Furstenberg, Y. Katznelson, \emph{A density version of the Hales-Jewett theorem for $k = 3$}, Graph Theory and Combinatorics (Cambridge, 1988). Discrete Math. 75 (1989), no. 1-3, 227–-241. | \bibitem{fk1} H. Furstenberg, Y. Katznelson, \emph{A density version of the Hales-Jewett theorem for $k = 3$}, Graph Theory and Combinatorics (Cambridge, 1988). Discrete Math. 75 (1989), no. 1-3, 227–-241. | ||

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\bibitem{Krisha} K. Krishna, M. Narasimha Murty, "Genetic K-means algorithm," Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on , vol.29, no.3, pp.433-439, Jun 1999 | \bibitem{Krisha} K. Krishna, M. Narasimha Murty, "Genetic K-means algorithm," Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on , vol.29, no.3, pp.433-439, Jun 1999 | ||

− | \bibitem{markstrom} {{\tt http://abel.math.umu.se/~klasm/Data/HJ/}} | + | \bibitem{markstrom} K. Markstrom, {{\tt http://abel.math.umu.se/~klasm/Data/HJ/}} |

\bibitem{moser} L. Moser, Problem P.170 in Canad. Math. Bull. 13 (1970), 268. | \bibitem{moser} L. Moser, Problem P.170 in Canad. Math. Bull. 13 (1970), 268. | ||

− | \bibitem{mcc} R. McCutcheon, \emph{The conclusion of the proof of the density Hales-Jewett theorem for k=3}, unpublished. | + | \bibitem{mcc} R. McCutcheon, \emph{The conclusion of the proof of the density Hales-Jewett theorem for $k=3$}, unpublished. |

\bibitem{obryant} | \bibitem{obryant} | ||

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\bibitem{oeis} | \bibitem{oeis} | ||

− | N. J. A. Sloane, Ed. (2008), The On-Line Encyclopedia of Integer Sequences, {\tt www.research.att.com/ | + | N. J. A. Sloane, Ed. (2008), The On-Line Encyclopedia of Integer Sequences, {\tt www.research.att.com/$\sim$njas/sequences/} |

\bibitem{potenchin} | \bibitem{potenchin} | ||

A. Potechin, \emph{Maximal caps in $AG(6, 3)$}, Journal Designs, Codes and Cryptography, Volume 46, Number 3 / March, 2008. | A. Potechin, \emph{Maximal caps in $AG(6, 3)$}, Journal Designs, Codes and Cryptography, Volume 46, Number 3 / March, 2008. | ||

− | \bibitem{poly} D.H.J. Polymath, | + | \bibitem{poly} D.H.J. Polymath, \emph{A new proof of the density Hales-Jewett theorem}, preprint, available at {\tt http://arxiv.org/abs/0910.3926}. |

+ | |||

+ | \bibitem{polywiki} D.H.J. Polymath, {\tt michaelnielsen.org/polymath1/index.php?title=Polymath1} | ||

\bibitem{rankin} | \bibitem{rankin} | ||

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\bibitem{szem} | \bibitem{szem} | ||

E. Szemer\'edi, \emph{On sets of integers containing no $k$ elements in arithmetic progression}, Acta Arithmetica \textbf{27} (1975), 199-–245. | E. Szemer\'edi, \emph{On sets of integers containing no $k$ elements in arithmetic progression}, Acta Arithmetica \textbf{27} (1975), 199-–245. | ||

− | |||

− | |||

\end{thebibliography} | \end{thebibliography} |

## Revision as of 06:39, 16 January 2010

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\begin{document}

\title{Density Hales-Jewett and Moser numbers}

\author{D.H.J. Polymath} \address{http://michaelnielsen.org/polymath1/index.php} %\email{}

\subjclass{05D05, 05D10}

\begin{abstract} For any $n \geq 0$ and $k \geq 1$, the \emph{density Hales-Jewett number} $c_{n,k}$ is defined as the size of the largest subset of the cube $[k]^n$ := $\{1,\ldots,k\}^n$ which contains no combinatorial line; similarly, the Moser number $c'_{n,k}$ is the largest subset of the cube $[k]^n$ which contains no geometric line. A deep theorem of Furstenberg and Katznelson \cite{fk1}, \cite{fk2}, \cite{mcc} shows that $c_{n,k}$ = $o(k^n)$ as $n \to \infty$ (which implies a similar claim for $c'_{n,k}$); this is already non-trivial for $k = 3$. Several new proofs of this result have also been recently established \cite{poly}, \cite{austin}.

Using both human and computer-assisted arguments, we compute several values of $c_{n,k}$ and $c'_{n,k}$ for small $n,k$. For instance the sequence $c_{n,3}$ for $n=0,\ldots,6$ is $1,2,6,18,52,150,450$, while the sequence $c'_{n,3}$ for $n=0,\ldots,6$ is $1,2,6,16,43,124,353$. We also establish some results for higher $k$, showing for instance that an analogue of the LYM inequality (which relates to the $k = 2$ case) does not hold for higher $k$. \end{abstract}

\maketitle %\today

%\setcounter{tocdepth}{1} %\tableofcontents

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \include{introduction} \include{dhj-lown-lower} \include{dhj-lown} \include{moser-lower} \include{moser} %\include{fujimura} %\include{higherk} % \include{coloring}

%\appendix

%\include{genetic} %\include{integer}

\begin{thebibliography}{10}

\bibitem{ajtai} M. Ajtai, E. Szemer\'edi, \emph{Sets of lattice points that form no squares}, Studia Scientiarum Mathematicarum Hungarica, 9, 9-11 (1974), (1975)

\bibitem{austin} T. Austin, \emph{Deducing the density Hales-Jewett theorem from an infinitary removal lemma}, preprint.

\bibitem{beck} J. Beck, Combinatorial Games: Tic-Tac-Toe Theory. Cambridge University Press, 2008.

\bibitem{behrend} F. Behrend, \emph{On the sets of integers which contain no three in arithmetic progression}, Proceedings of the National Academy of Sciences \textbf{23} (1946), 331–-332.

\bibitem{Brower} A. Brower, {\tt www.win.tue.nl/$\sim$aeb/codes/binary-1.html}.

\bibitem{chandra} A. Chandra, \emph{On the solution of Moser's problem in four dimensions}, Canad. Math. Bull. \textbf{16} (1973), 507--511.

\bibitem{chvatal1} V. Chv\'{a}tal, \emph{Remarks on a problem of Moser}, Canadian Math Bulletin, Vol 15, 1972, 19--21.

\bibitem{chvatal2} V. Chv\'{a}tal, \emph{Edmonds polytopes and a hierarchy of combinatorial problems}, Discrete Math. 4 (1973) 305-337.

\bibitem{elkin} M. Elkin, \emph{An Improved Construction of Progression-Free Sets}, preprint.

\bibitem{fuji} K. Fujimura, {\tt www.puzzles.com/PuzzlePlayground/CoinsAndTriangles/CoinsAndTriangles.htm}

\bibitem{fk1} H. Furstenberg, Y. Katznelson, \emph{A density version of the Hales-Jewett theorem for $k = 3$}, Graph Theory and Combinatorics (Cambridge, 1988). Discrete Math. 75 (1989), no. 1-3, 227–-241.

\bibitem{fk2} H. Furstenberg, Y. Katznelson, \emph{A density version of the Hales-Jewett theorem}, J. Anal. Math. 57 (1991), 64–-119. MR1191743

\bibitem{greenwolf} B. Green, J. Wolf, \emph{A note on Elkin's improvement of Behrend's construction}, preprint.

\bibitem{heule} Marijn Heule, presentation at {\tt http://www.st.ewi.tudelft.nl/sat/slides/waerden.pdf}

\bibitem{komlos} J. Koml\'{o}s, solution to problem P.170 by Leo Moser, Canad. Math.. Bull. vol {\bf (??check)} (1972), 312--313, 1970.

\bibitem{Krisha} K. Krishna, M. Narasimha Murty, "Genetic K-means algorithm," Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on , vol.29, no.3, pp.433-439, Jun 1999

\bibitem{markstrom} K. Markstrom, Template:\tt http://abel.math.umu.se/~klasm/Data/HJ/

\bibitem{moser} L. Moser, Problem P.170 in Canad. Math. Bull. 13 (1970), 268.

\bibitem{mcc} R. McCutcheon, \emph{The conclusion of the proof of the density Hales-Jewett theorem for $k=3$}, unpublished.

\bibitem{obryant} K. O'Bryant, \emph{Sets of integers that do not contain long arithmetic progressions}, preprint.

\bibitem{oeis} N. J. A. Sloane, Ed. (2008), The On-Line Encyclopedia of Integer Sequences, {\tt www.research.att.com/$\sim$njas/sequences/}

\bibitem{potenchin} A. Potechin, \emph{Maximal caps in $AG(6, 3)$}, Journal Designs, Codes and Cryptography, Volume 46, Number 3 / March, 2008.

\bibitem{poly} D.H.J. Polymath, \emph{A new proof of the density Hales-Jewett theorem}, preprint, available at {\tt http://arxiv.org/abs/0910.3926}.

\bibitem{polywiki} D.H.J. Polymath, {\tt michaelnielsen.org/polymath1/index.php?title=Polymath1}

\bibitem{rankin} R. A. Rankin, Sets of integers containing not more than a given number of terms in arithmetical progression, Proc. Roy. Soc. Edinburgh Sect. A 65 (1960/1961), 332–344 (1960/61). MR 0142526 (26 \#95)

\bibitem{roth} K. Roth, \emph{On certain sets of integers, I}, Journal of the London Mathematical Society \textbf{28} (1953), 104-–109.

\bibitem{Rothlauf} F. Rothlauf, D. E. Goldberg, Representations for Genetic and Evolutionary Algorithms. Physica-Verlag, 2002.

\bibitem{shelah} S. Shelah, \emph{Primitive recursive bounds for van der Warden numbers}, Journal of the American Mathematical Society \textbf{28} 1988, 683-–697.

\bibitem{sperner} E. Sperner, \emph{Ein Satz \"uber Untermengen einer endlichen Menge}, Mathematische Zeitschrift \textbf{27} (1928), 544-–548.

\bibitem{szem} E. Szemer\'edi, \emph{On sets of integers containing no $k$ elements in arithmetic progression}, Acta Arithmetica \textbf{27} (1975), 199-–245.

\end{thebibliography}

\end{document}