Open Problem Garden - Rainbow AP(4) in an almost equinumerous coloring - Comments http://openproblemgarden.org/op/rainbow_ap_4_in_an_almost_equinumerous_coloring Comments for "Rainbow AP(4) in an almost equinumerous coloring" en Tight hypergraphs (re: Rainbow AP(4) in an almost equinumerous coloring) http://openproblemgarden.org/op/rainbow_ap_4_in_an_almost_equinumerous_coloring#comment-160 <p>It deservs to be mentioned that in any <img class="teximage" src="/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png" alt="$ 3 $" />-colouring of <img class="teximage" src="/files/tex/5acb62d24a3554bd830b4b480f7d1046c6d49235.png" alt="$ {\mathbb Z}_p^*/{\mathbb Z}_3^* $" />, the equation <img class="teximage" src="/files/tex/1e28bbdd9b1366e6fcc226d5141e6eeafba08fab.png" alt="$ x+y=z $" />, have an heterochromatic (rainbow) solution; here, <img class="teximage" src="/files/tex/137c9ce3d6c6277fa2edf563bcc0c4583d89a49f.png" alt="$ {\mathbb Z}_p^* $" /> denotes the multiplicative group of the field <img class="teximage" src="/files/tex/e8c94ceb5a9d688bff114c12f7fe9fe47ef955fc.png" alt="$ {\mathbb Z}_p $" />.</p> Sat, 01 Sep 2007 23:32:58 +0200 Dino comment 160 at http://openproblemgarden.org Rainbow AP(4) in an almost equinumerous coloring http://openproblemgarden.org/op/rainbow_ap_4_in_an_almost_equinumerous_coloring <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/conlon_david">Conlon</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/combinatorics">Combinatorics</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <div class="envtheorem"><b>Problem</b>&nbsp;&nbsp; Do 4-colorings of <img class="teximage" src="/files/tex/ca03dcde1fc73a1d3d1916bca138cd11161ea69a.png" alt="$ \mathbb{Z}_{p} $" />, for <img class="teximage" src="/files/tex/928cd9d544fdea62f88a627aaee28c416c4366c0.png" alt="$ p $" /> a large prime, always contain a rainbow <img class="teximage" src="/files/tex/abd6fa4428454b30450d94292e000c9ecaa7a4fc.png" alt="$ AP(4) $" /> if each of the color classes is of size of either <img class="teximage" src="/files/tex/d906d60e3b848dd93ec8be196b73063411f71d25.png" alt="$ \lfloor p/4\rfloor $" /> or <img class="teximage" src="/files/tex/5e7562752899a961fe9ccacfd0a84316504a88c2.png" alt="$ \lceil p/4\rceil $" />? </div> </tr></td> </table> </td> </tr> </table> Conlon, David arithmetic progression rainbow Combinatorics http://openproblemgarden.org/op/rainbow_ap_4_in_an_almost_equinumerous_coloring#comment Fri, 20 Jul 2007 20:10:19 +0200 vjungic 478 at http://openproblemgarden.org