http://openproblemgarden.org/op/geometric_hales_jewett_theorem Comments for "Geometric Hales-Jewett Theorem" en http://openproblemgarden.org/op/geometric_hales_jewett_theorem <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/por_attila">Por</a>; <a href="/category/wood_david_r">Wood</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/geometry">Geometry</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; For all integers <img class="teximage" src="/files/tex/636eac0b23e9e5ba1f1a1fc5e22e2d2009ff1533.png" alt="$ k\geq1 $" /> and <img class="teximage" src="/files/tex/06509000d5e4893e990a6bf4e2deca4af4e82a6c.png" alt="$ \ell\geq3 $" />, there is an integer <img class="teximage" src="/files/tex/b9ec548be74b7eb11cfc1d7f1b688bc508002543.png" alt="$ f(k,\ell) $" /> such that for every set <img class="teximage" src="/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png" alt="$ P $" /> of at least <img class="teximage" src="/files/tex/b9ec548be74b7eb11cfc1d7f1b688bc508002543.png" alt="$ f(k,\ell) $" /> points in the plane, if each point in <img class="teximage" src="/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png" alt="$ P $" /> is assigned one of <img class="teximage" src="/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png" alt="$ k $" /> colours, then:<br /> <ul class="itemize"> \item <img class="teximage" src="/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png" alt="$ P $" /> contains <img class="teximage" src="/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png" alt="$ \ell $" /> collinear points, or \item <img class="teximage" src="/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png" alt="$ P $" /> contains a monochromatic line (that is, a maximal set of collinear points receiving the same colour) </ul> </div> </tr></td> </table> </td> </tr> </table> Por, Attila Wood, David R. Hales-Jewett Theorem ramsey theory Geometry http://openproblemgarden.org/op/geometric_hales_jewett_theorem#comment Wed, 26 Mar 2014 18:07:18 +0100 David Wood 59925 at http://openproblemgarden.org