Given a set family, what is the best way (empirically) to check whether the set family is equivalent to set of independent sets of some matroid. The input can be either the set family explicitly or bunch of cardinality constraints that must be satisfied. For example, given a universe E, subsets $A\_1,..,A\_k$ of E and positive integers $b\_1,...,b\_k$, a set F is in the family iff $|F\cap A\_i|<= b\_i$ for each $1<= i<= k$. Check whether the family is a matroid.



Checking whether a set family forms a matroid.
One way is to delete an element $x$ from your alleged matroid $M$, recursively check that the smaller structure $M'$ is a matroid, and then check that adding back $x$ gives you a single-element extension of $M'$. An algorithm for testing for single-element extensions is explained in [this paper by Mayhew and Royle](http://arxiv.org/abs/math/0702316), in which they compute all matroids with up to nine elements.


