http://www.openproblemgarden.org/op/concavity_of_van_der_waerden_numbers Comments for "Concavity of van der Waerden numbers" en http://www.openproblemgarden.org/op/concavity_of_van_der_waerden_numbers <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/landman_bruce">Landman</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/combinatorics">Combinatorics</a> » <a href="/category/ramsey_theory">Ramsey Theory</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>For <img class="teximage" src="/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png" alt="$ k $" /> and <img class="teximage" src="/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png" alt="$ \ell $" /> positive integers, the (mixed) van der Waerden number <img class="teximage" src="/files/tex/291c9d91508f6d64f719f93de43833244c947faa.png" alt="$ w(k,\ell) $" /> is the least positive integer <img class="teximage" src="/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png" alt="$ n $" /> such that every (red-blue)-coloring of <img class="teximage" src="/files/tex/661d0acc09fbc5b62f49645a71cf7d831a26563f.png" alt="$ [1,n] $" /> admits either a <img class="teximage" src="/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png" alt="$ k $" />-term red arithmetic progression or an <img class="teximage" src="/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png" alt="$ \ell $" />-term blue arithmetic progression. </p> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; For all <img class="teximage" src="/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png" alt="$ k $" /> and <img class="teximage" src="/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png" alt="$ \ell $" /> with <img class="teximage" src="/files/tex/739b55f8a55e492f6a853fe336e2a04b05bd1182.png" alt="$ k \geq \ell $" />, <img class="teximage" src="/files/tex/783e69162d75cf0eb2d5bd5a8d8b3410b04e5107.png" alt="$ w(k,\ell) \geq w(k+1,\ell-1) $" />. </div> </tr></td> </table> </td> </tr> </table> Landman, Bruce M. arithmetic progression van der Waerden Combinatorics Ramsey Theory http://www.openproblemgarden.org/op/concavity_of_van_der_waerden_numbers#comment Thu, 21 Jun 2007 19:59:29 +0200 Bruce Landman 404 at http://www.openproblemgarden.org