Rota conjectured that if $(B_1,\ldots,B_n)$ are disjoint bases in a rank-n matroid M, then there are n disjoint transversals of $(B_1,\ldots,B_n)$ that are bases of M. We prove the weaker result that there are $O(\sqrt n)$ disjoint transversals of $(B_1,\ldots,B_n)$ that are bases. We also prove that if $(B_1,\ldots,B_k)$ are disjoint bases of a rank-n matroid with $n> \binom{k+1}{2}$, then there are n disjoint independent transversals of $(B_1,\ldots,B_k)$.

We introduce a new generalization of the maximum matching problem to matroids; this problem includes Gallai’s T‐path problem for graphs.

Rota conjectured that, given n disjoint bases of a rank‐n matroid M, there are n disjoint transversals of these bases that are all bases of M. We prove a stronger statement for the class of paving matroids.

We prove that if N is an internally 4‐connected minor of an internally 4‐connected binary matroid M with $E(N) \geq 4$, then there exist matroids $M_0, M_1, \ldots, M_n$ such that $M_0 \cong N$, $M_n = M$, and, for each $i\in\{1,\ldots,i\}$, $M_{i-1}$ is a minor of $M_{i}$, $|E(M_{i-1})|\ge |E(M_i)|-2$, and $M_i$ is 4‐connected up to separators of size 5.

Let (X 1 ,X 2 ) be an exact k-separation of a matroid N. If M is a matroid that contains N as a minor and the k-separation (X 1 ,X 2 ) does not extend to a k-separation in M, then we say that Mbridges the k-separation (X 1 ,X 2 ) in N. One would hope that a minor minimal bridge for (X 1 ,X 2 ) would not be much larger than N. Unfortunately there are instances in which one can construct arbitrarily large minor-minimal bridges. We restrict our attention to the class of matroids representable over a fixed finite field and show that here minor-minimal bridges are bounded in size.

We describe a common generalization of the weighted matching problem and the weighted matroid intersection problem. In this context we present results implying the polynomial-time solvability of the two problems. We also use our results to give the first strongly polynomial separation algorithm for the convex hull of matchable sets of a graph, and the first polynomial-time algorithm to compute the rank of a certain matrix of indeterminates. Our algorithmic results are based on polyhedral characterizations, and on the equivalence of separation and optimization.