I'm concerned with the following problem:


We are given a set $\mathcal{A} \subseteq 2^S$ of subsets of the ground set $S=E(M)$ of a matroid $M$. I would like to know whether there is a basis $B$ of the matroid which intersects every set in $A$, meaning $B \cap A \neq \emptyset , \forall A \in \mathcal{A}.$ 
Is there a simple criterion for deciding this question? With "simple" I don't necessarily mean effective or fast, but much more mathematically nice, like e.g. the rado/hall conditions for the existence of transversals with special properties.


Thank you.



Criterion for the existence of a basis in a matroid intersecting given sets
This is an elaboration of Fedor Petrov's comment. Suppose we could give a satisfactory answer to your question for uniform matroids. Then we would have a nice way to characterize solutions to the hitting set problem (which asks for the size of the smallest set that intersects every set in $\mathcal{A}$). But the hitting set problem is NP-hard. While this doesn't prove that there couldn't be a nice characterization, I'm not aware of any NP-hard problems where the satisfiable instances have a nice characterization (by which I mean a simple and non-trivial reformulation of the original definition of the problem). Recall for example that Hall's theorem comes along with a polynomial time algorithm.


