http://openproblemgarden.org/op/long_rainbow_arithemtic_progressions Comments for "Long rainbow arithmetic progressions" en http://openproblemgarden.org/op/long_rainbow_arithemtic_progressions#comment-6729 <p>Looks like a nice problem, but is it still unsolved?</p> Wed, 12 May 2010 05:59:22 +0200 Anonymous comment 6729 at http://openproblemgarden.org http://openproblemgarden.org/op/long_rainbow_arithemtic_progressions <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/fox_jacob">Fox</a>; <a href="/category/jungic_veselin">Jungic</a>; <a href="/category/mahdian_mohammad">Mahdian</a>; <a href="/category/nesetril">Nesetril</a>; <a href="/category/radoicic_rados">Radoicic</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/combinatorics">Combinatorics</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>For <img class="teximage" src="/files/tex/c46fca02415dc26dddad26ae8da6b1d9ba66ecb8.png" alt="$ k\in \mathbb{N} $" /> let <img class="teximage" src="/files/tex/2e5b316262d50b60e8c018aa016d79d00ad8d3dd.png" alt="$ T_k $" /> denote the minimal number <img class="teximage" src="/files/tex/94f50738cb479379b60bd3be273833fbb031f8e5.png" alt="$ t\in \mathbb{N} $" /> such that there is a rainbow <img class="teximage" src="/files/tex/d7238e431190a14e4a1d8e4c83517c9c67f3117a.png" alt="$ AP(k) $" /> in every equinumerous <img class="teximage" src="/files/tex/4761b031c89840e8cd2cda5b53fbc90c308530f3.png" alt="$ t $" />-coloring of <img class="teximage" src="/files/tex/ed1bb6b613e82aadd4a130826f1c88269edeb79b.png" alt="$ \{ 1,2,\ldots ,tn\} $" /> for every <img class="teximage" src="/files/tex/486c9aba0fe5968add271676a60f73108872b6c2.png" alt="$ n\in \mathbb{N} $" /></p> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; For all <img class="teximage" src="/files/tex/c403b537b372cb5d16b6129dce0f0c416c1f1b31.png" alt="$ k\geq 3 $" />, <img class="teximage" src="/files/tex/861c6fb3fc588a2247540bcd7bf32e42032ad265.png" alt="$ T_k=\Theta (k^2) $" />. </div> </tr></td> </table> </td> </tr> </table> Fox, Jacob Jungic, Veselin Mahdian, Mohammad Nesetril, Jaroslav Radoicic, Rados arithmetic progression rainbow Combinatorics http://openproblemgarden.org/op/long_rainbow_arithemtic_progressions#comment Tue, 03 Jul 2007 04:14:18 +0200 vjungic 426 at http://openproblemgarden.org