http://openproblemgarden.org/op/aharoni_berger_conjecture Comments for "Aharoni-Berger conjecture" en http://openproblemgarden.org/op/aharoni_berger_conjecture <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/aharoni">Aharoni</a>; <a href="/category/berger_eli">Berger</a>&nbsp;&nbsp; </td> <td align=right> Subject: <a href="/category/combinatorics">Combinatorics</a> » <a href="/category/matroid_theory">Matroid 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; If <img class="teximage" src="/files/tex/368dea3f4a89576f8e4eebf3241a6ef062e5b5d9.png" alt="$ M_1,\ldots,M_k $" /> are matroids on <img class="teximage" src="/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png" alt="$ E $" /> and <img class="teximage" src="/files/tex/d896134dc1e4119543db8e0baaebe50b9bb34085.png" alt="$ \sum_{i=1}^k rk_{M_i}(X_i) \ge \ell (k-1) $" /> for every partition <img class="teximage" src="/files/tex/af99ea0d6ceb5907ffb549d85a7c7e711c6b91c7.png" alt="$ \{X_1,\ldots,X_k\} $" /> of <img class="teximage" src="/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png" alt="$ E $" />, then there exists <img class="teximage" src="/files/tex/3b78a133c1f7e4ab67c2c7c5a7fbba992d13e9dc.png" alt="$ X \subseteq E $" /> with <img class="teximage" src="/files/tex/6ed3addf64c5678ff6cd1add3575f29c9a04af48.png" alt="$ |X| = \ell $" /> which is independent in every <img class="teximage" src="/files/tex/71c6239abfbaf80e551a379622380b245aaae23a.png" alt="$ M_i $" />. </div> </tr></td> </table> </td> </tr> </table> Aharoni, Ron Berger, Eli independent set matroid partition Combinatorics Matroid Theory http://openproblemgarden.org/op/aharoni_berger_conjecture#comment Mon, 11 Jun 2007 05:40:27 +0200 mdevos 382 at http://openproblemgarden.org