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matroid
Aharoni-Berger conjecture ★★★
Conjecture If
are matroids on
and
for every partition
of
, then there exists
with
which is independent in every
.
![$ M_1,\ldots,M_k $](/files/tex/368dea3f4a89576f8e4eebf3241a6ef062e5b5d9.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ \sum_{i=1}^k rk_{M_i}(X_i) \ge \ell (k-1) $](/files/tex/d896134dc1e4119543db8e0baaebe50b9bb34085.png)
![$ \{X_1,\ldots,X_k\} $](/files/tex/af99ea0d6ceb5907ffb549d85a7c7e711c6b91c7.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ X \subseteq E $](/files/tex/3b78a133c1f7e4ab67c2c7c5a7fbba992d13e9dc.png)
![$ |X| = \ell $](/files/tex/6ed3addf64c5678ff6cd1add3575f29c9a04af48.png)
![$ M_i $](/files/tex/71c6239abfbaf80e551a379622380b245aaae23a.png)
Keywords: independent set; matroid; partition
Bases of many weights ★★★
Let be an (additive) abelian group, and for every
let
.
Conjecture Let
be a matroid on
, let
be a map, put
and
. Then
![$ M $](/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ w : E \rightarrow G $](/files/tex/7c1a9b6ba2a67b002e25339803f3d5a6da1b684d.png)
![$ S = \{ \sum_{b \in B} w(b) : B \mbox{ is a base} \} $](/files/tex/15b6767566360480b09b1982f902c978265de9c9.png)
![$ H = {\mathit stab}(S) $](/files/tex/e338cf1ff9295e47c3f1552771eb4fecfb4d730f.png)
![$$|S| \ge |H| \left( 1 - rk(M) + \sum_{Q \in G/H} rk(w^{-1}(Q)) \right).$$](/files/tex/e22b2f4b60bf6bbfd304dd219b3092fb50cbae67.png)
Rota's unimodal conjecture ★★★
Author(s): Rota
Let be a matroid of rank
, and for
let
be the number of closed sets of rank
.
Conjecture
is unimodal.
![$ w_0,w_1,\ldots,w_r $](/files/tex/44d0d863f1306476a87d39f7b389e116afd6410d.png)
Conjecture
is log-concave.
![$ w_0,w_1,\ldots,w_r $](/files/tex/44d0d863f1306476a87d39f7b389e116afd6410d.png)
Keywords: flat; log-concave; matroid
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