We show that given two bases $A$ and $B$ of a matroid $\mathcal M$ and any given partition $\bigcup_{i=1}^kB_i$ of $B$ there exists a partition $\bigcup_{i=1}^kA_i$ of $A$ such that $(A- A_i)\cup B_i$ and $(A- \bigcup_{j=1}^i A_j)\cup (\bigcup_{j=1}^i B_j)$ are bases for all $i=1,\ldots,k$. We also give a new proof, not relying on Hall's theorem, of a theorem of Donald and Tobey stating that for each $k=1,\dots,n$, there exists a bijection $\tau_k$ from $k$-subsets $I$ of $A$ to $k$-subsets of $B$ such that $(A- I)\cup\tau_k(I)$ is always a basis.

Let $f(n)$ be the smallest number such that every collection of $n$ matchings, each of size at least $f(n)$, in a bipartite graph, has a full rainbow matching. Generalizing famous conjectures of Ryser, Brualdi, and Stein, Aharoni and Berger [ Electron. J. Combin., 16 (2009), R119] conjectured that $f(n)=n+1$ for every $n>1$. Clemens and Ehrenmüller proved that $f(n) \le \frac{3}{2}n +o(n)$. We show that the $o(n)$ term can be reduced to a constant, namely, $f(n) \le \lceil \frac{3}{2}n \rceil+1$.

The Alon--Tarsi Latin squares conjecture is extended to odd dimensions by stating it for reduced Latin squares (Latin squares having the identity permutation as their first row and first column). Using a modified version of an identity proved by Onn [Amer. Math. Monthly, 104 (1997), pp. 156--159], we show that the validity of this conjecture implies a weak version of Rota's bases conjecture for odd dimensions, namely that a set of $n$ bases in $\mathbb{R}^n$ has $n-1$ disjoint independent transversals.

The way circuits, relative to a basis, are affected as a result of exchanging a basis element is studied. As consequences, it is shown that three consecutive symmetric exchanges exist for any two bases of a matroid, and that a full serial symmetric exchange, of length at most 6, exists for any two bases of a matroid of rank 5. A new characterization of binary matroids, related to basis-exchange, is presented.