http://openproblemgarden.org/op/bases_of_many_weights Comments for "Bases of many weights" en http://openproblemgarden.org/op/bases_of_many_weights <table cellspacing="10"> <tr> <td> Author(s): <a href="/category/schrijver_alexander">Schrijver</a>; <a href="/category/seymour">Seymour</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> <p>Let <img class="teximage" src="/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png" alt="$ G $" /> be an (additive) abelian group, and for every <img class="teximage" src="/files/tex/1443dce39cbce9606ca2c26bfb4ffbcf28ddb2c9.png" alt="$ S \subseteq G $" /> let <img class="teximage" src="/files/tex/2fab424b5457f67d5c6c3f4d4e3d04269eb27ed9.png" alt="$ {\mathit stab}(S) = \{ g \in G : g + S = S \} $" />. </p> <div class="envtheorem"><b>Conjecture</b>&nbsp;&nbsp; Let <img class="teximage" src="/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png" alt="$ M $" /> be a matroid on <img class="teximage" src="/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png" alt="$ E $" />, let <img class="teximage" src="/files/tex/7c1a9b6ba2a67b002e25339803f3d5a6da1b684d.png" alt="$ w : E \rightarrow G $" /> be a map, put <img class="teximage" src="/files/tex/15b6767566360480b09b1982f902c978265de9c9.png" alt="$ S = \{ \sum_{b \in B} w(b) : B \mbox{ is a base} \} $" /> and <img class="teximage" src="/files/tex/e338cf1ff9295e47c3f1552771eb4fecfb4d730f.png" alt="$ H = {\mathit stab}(S) $" />. Then <img class="teximage" src="/files/tex/e22b2f4b60bf6bbfd304dd219b3092fb50cbae67.png" alt="$$|S| \ge |H| \left( 1 - rk(M) + \sum_{Q \in G/H} rk(w^{-1}(Q)) \right).$$" /> </div> </tr></td> </table> </td> </tr> </table> Schrijver, Alexander Seymour, Paul D. matroid sumset zero sum Combinatorics Matroid Theory http://openproblemgarden.org/op/bases_of_many_weights#comment Sun, 10 Jun 2007 03:51:25 +0200 mdevos 369 at http://openproblemgarden.org