http://openproblemgarden.org/op/graphs_of_exact_colorings Comments for "Graphs of exact colorings" en http://openproblemgarden.org/op/graphs_of_exact_colorings <table cellspacing="10"> <tr> <td> Author(s): <a href="/kpz_equation_central_limit_theorem"></a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/algebra">Algebra</a>&nbsp;&nbsp; </td> </tr> <tr> <td colspan=2> <table border=1 cellspacing="5"> <tr><td> <p>Conjecture For <img class="teximage" src="/files/tex/5b78c1e3ea22284a95651ba44f108994d6cf2f86.png" alt="$ c \geq m \geq 1 $" />, let <img class="teximage" src="/files/tex/cfe5c589cc4a2226974cf81921f4d1d715069067.png" alt="$ P(c,m) $" /> be the statement that given any exact <img class="teximage" src="/files/tex/ddf676895ab4ddd0cd7af3c302b85a1f60adaad2.png" alt="$ c $" />-coloring of the edges of a complete countably infinite graph (that is, a coloring with <img class="teximage" src="/files/tex/ddf676895ab4ddd0cd7af3c302b85a1f60adaad2.png" alt="$ c $" /> colors all of which must be used at least once), there exists an exactly <img class="teximage" src="/files/tex/55bdca2ca0a9423bb71d3442ea8588cb8365484d.png" alt="$ m $" />-colored countably infinite complete subgraph. Then <img class="teximage" src="/files/tex/cfe5c589cc4a2226974cf81921f4d1d715069067.png" alt="$ P(c,m) $" /> is true if and only if <img class="teximage" src="/files/tex/8e814d3d1e3c4b23b843ccf5d4d6cc8b88c8138e.png" alt="$ m=1 $" />, <img class="teximage" src="/files/tex/b1b76246cf7af2dbfff6ad35b47748ace2fb112c.png" alt="$ m=2 $" />, or <img class="teximage" src="/files/tex/85606e9d4efe762d2d9f5f02e4ae76fb8200e2f7.png" alt="$ c=m $" />.</p> </tr></td> </table> </td> </tr> </table> Algebra http://openproblemgarden.org/op/graphs_of_exact_colorings#comment Thu, 03 Oct 2013 18:25:03 +0200 sabisood 57824 at http://openproblemgarden.org