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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}. | 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 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 $ | 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} | ||
| Line 85: | Line 85: | ||
\begin{thebibliography}{10} | \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 | \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. | \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. | \bibitem{beck} J. Beck, Combinatorial Games: Tic-Tac-Toe Theory. Cambridge University Press, 2008, Cambridge. | ||
\bibitem{behrend} | \bibitem{behrend} | ||
| Line 100: | Line 100: | ||
A. Chandra, \emph{On the solution of Moser's problem in four dimensions}, Canad. Math. Bull. \textbf{16} (1973), 507--511. | 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}, | \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. 4 (1973) 305-337. | \bibitem{chvatal2} V. Chv\'{a}tal, \emph{Edmonds polytopes and a hierarchy of combinatorial problems}, Discrete Math. \textbf{4} (1973) 305--337. | ||
\bibitem{elkin} | \bibitem{elkin} | ||
| Line 110: | Line 110: | ||
K. Fujimura, {\tt www.puzzles.com/PuzzlePlayground/CoinsAndTriangles/CoinsAndTriangles.htm} | 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) | \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. 57 (1991), 64–-119. | \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} | \bibitem{kra} | ||
D. Geller, I. Kra, S. Popescu, S. Simanca, \emph{On circulant matrices}, {\tt www.math.sunysb.edu/ | D. Geller, I. Kra, S. Popescu, S. Simanca, \emph{On circulant matrices}, {\tt www.math.sunysb.edu/$\sim$sorin/eprints/circulant.pdf} | ||
\bibitem{greenwolf} | \bibitem{greenwolf} | ||
B. Green, J. Wolf, \emph{A note on Elkin's improvement of Behrend's construction}, preprint. | 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} | \bibitem{heule} M. Heule, presentation at {\tt www.st.ewi.tudelft.nl/sat/slides/waerden.pdf} | ||
\bibitem{komlos} | \bibitem{komlos} | ||
J. Koml\'{o}s, solution to problem P.170 by Leo Moser, Canad. Math | 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{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 | ||
| Line 129: | Line 129: | ||
\bibitem{markstrom} K. Markstrom, {{\tt abel.math.umu.se/$\sim$klasm/Data/HJ/}} | \bibitem{markstrom} K. Markstrom, {{\tt abel.math.umu.se/$\sim$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. \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{mcc} R. McCutcheon, \emph{The conclusion of the proof of the density Hales-Jewett theorem for $k=3$}, unpublished. | ||
\bibitem{obryant} | \bibitem{obryant} | ||
K. O'Bryant, \emph{Sets of integers that do not contain long arithmetic progressions}, preprint. | 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} | \bibitem{oeis} | ||
| Line 140: | Line 140: | ||
\bibitem{potenchin} | \bibitem{potenchin} | ||
A. Potechin, \emph{Maximal caps in $AG(6, 3)$}, | 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{poly} D.H.J. Polymath, \emph{A new proof of the density Hales-Jewett theorem}, preprint, available at {\tt arxiv.org/abs/0910.3926}. | ||
| Line 147: | Line 147: | ||
\bibitem{rankin} | \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), | 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} | \bibitem{roth} | ||
K. Roth, \emph{On certain sets of integers, I}, | 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}, | \bibitem{shelah} S. Shelah, \emph{Primitive recursive bounds for van der Warden numbers}, J. Amer. Math. Soc. \textbf{28} (1988), 683-–697. | ||
\bibitem{sperner} | \bibitem{sperner} | ||
Latest revision as of 19:37, 25 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 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}
