# Difference between revisions of "Polymath.tex"

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− | \title{Density Hales-Jewett and Moser numbers | + | \title{Density Hales-Jewett and Moser numbers} |

\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, | + | 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 prove 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 $k$, and also establishing the asymptotic lower bound $c_{n,k} \geq k^n \exp\left( - O(\sqrt[\ell]{\log n})\right)$ where $\ell$ is the largest integer such that $2k > 2^\ell$. |

\end{abstract} | \end{abstract} | ||

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%\today | %\today | ||

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

− | \tableofcontents | + | %\tableofcontents |

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

− | \ | + | \input{introduction} |

+ | \input{dhj-lown-lower} | ||

+ | \input{dhj-lown} | ||

+ | \input{moser-lower} | ||

+ | \input{moser} | ||

+ | %\input{fujimura} | ||

+ | %\input{higherk} | ||

+ | % \input{coloring} | ||

− | + | %\appendix | |

+ | %\input{genetic} | ||

+ | %\input{integer} | ||

− | + | \begin{thebibliography}{10} | |

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

− | + | ||

− | \ | + | \bibitem{austin} T. Austin, \emph{Deducing the density Hales-Jewett theorem from an infinitary removal lemma}, preprint, available at {\tt arxiv.org/abs/0903.1633}. |

− | + | ||

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

− | . | + | \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}, Canad. Math. Bull., \textbf{15} (1972) 19--21. |

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

− | + | ||

− | \ | + | \bibitem{elkin} |

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

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− | + | \bibitem{fuji} | |

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

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− | + | \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. \textbf{75} (1989), 227–-241. | |

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

− | \ | + | \bibitem{kra} |

+ | D. Geller, I. Kra, S. Popescu, S. Simanca, \emph{On circulant matrices}, {\tt www.math.sunysb.edu/$\sim$sorin/eprints/circulant.pdf} | ||

− | \ | + | \bibitem{greenwolf} |

+ | B. Green, J. Wolf, \emph{A note on Elkin's improvement of Behrend's construction}, preprint, available at {\tt arxiv.org/abs/0810.0732}. | ||

− | \ | + | \bibitem{heule} M. Heule, presentation at {\tt 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. \textbf{15} (1972), 312--313, 1970. | ||

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

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

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

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

− | \bibitem{ | + | \bibitem{obryant} |

+ | K. O'Bryant, \emph{Sets of integers that do not contain long arithmetic progressions}, preprint, available at {\tt arxiv.org/abs/0811.3057}. | ||

− | \bibitem{ | + | \bibitem{oeis} |

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

− | \bibitem{ | + | \bibitem{potenchin} |

− | + | A. Potechin, \emph{Maximal caps in $AG(6, 3)$}, Des. Codes Cryptogr., \textbf{46} (2008), 243--259. | |

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

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

− | \bibitem{ | + | \bibitem{rankin} |

+ | R. A. Rankin, \emph{Sets of integers containing not more than a given number of terms in arithmetical progression}, Proc. Roy. Soc. Edinburgh Sect. A \textbf{65} (1960/1961), 332–-344. | ||

− | \bibitem{ | + | \bibitem{roth} |

+ | K. Roth, \emph{On certain sets of integers, I}, J. Lond. Math. Soc. \textbf{28} (1953), 104-–109. | ||

− | \bibitem{Rothlauf} F. Rothlauf, D. E. Goldberg, Representations for Genetic and Evolutionary Algorithms. Physica-Verlag, 2002. | + | %\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}, J. Amer. Math. Soc. \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{thebibliography} |

## Latest revision as of 03:37, 26 January 2010

\documentclass[12pt,a4paper,reqno]{amsart} \usepackage{amssymb} \usepackage{amscd} %\usepackage{psfig} \usepackage{graphicx} %\usepackage{showkeys} % uncomment this when editing cross-references \numberwithin{equation}{section}

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\theoremstyle{plain}

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\theoremstyle{definition}

%\newtheorem{roughdef}[subsection]{Rough Definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{remarks}[theorem]{Remarks} \newtheorem{example}[theorem]{Example} \newtheorem{examples}[theorem]{Examples} %\newtheorem{problem}[subsection]{Problem} %\newtheorem{question}[subsection]{Question}

\newcommand\E{\mathbb{E}} \newcommand\R{\mathbb{R}} \newcommand\Z{\mathbb{Z}} \newcommand\N{\mathbb{N}} \renewcommand\P{\mathbb{P}}

\parindent 0mm \parskip 5mm

\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 prove 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 $k$, and also establishing the asymptotic lower bound $c_{n,k} \geq k^n \exp\left( - O(\sqrt[\ell]{\log n})\right)$ where $\ell$ is the largest integer such that $2k > 2^\ell$. \end{abstract}

\maketitle %\today

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

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

%\appendix

%\input{genetic} %\input{integer}

\begin{thebibliography}{10}

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

\bibitem{austin} T. Austin, \emph{Deducing the density Hales-Jewett theorem from an infinitary removal lemma}, preprint, available at {\tt arxiv.org/abs/0903.1633}.

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

\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}, Canad. Math. Bull., \textbf{15} (1972) 19--21.

\bibitem{chvatal2} V. Chv\'{a}tal, \emph{Edmonds polytopes and a hierarchy of combinatorial problems}, Discrete Math. \textbf{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. \textbf{75} (1989), 227–-241.

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

\bibitem{kra} D. Geller, I. Kra, S. Popescu, S. Simanca, \emph{On circulant matrices}, {\tt www.math.sunysb.edu/$\sim$sorin/eprints/circulant.pdf}

\bibitem{greenwolf} B. Green, J. Wolf, \emph{A note on Elkin's improvement of Behrend's construction}, preprint, available at {\tt arxiv.org/abs/0810.0732}.

\bibitem{heule} M. Heule, presentation at {\tt 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. \textbf{15} (1972), 312--313, 1970.

%\bibitem{Krisha} K. Krishna, M. Narasimha Murty, \emph{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 abel.math.umu.se/$\sim$klasm/Data/HJ/

\bibitem{moser} L. Moser, Problem P.170 in Canad. Math. Bull. \textbf{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, available at {\tt arxiv.org/abs/0811.3057}.

\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 $AG(6, 3)$}, Des. Codes Cryptogr., \textbf{46} (2008), 243--259.

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

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

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

\bibitem{roth} K. Roth, \emph{On certain sets of integers, I}, J. Lond. Math. Soc. \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}, J. Amer. Math. Soc. \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}