# Disjoint Common Transversals of Two Families of Sets

Let $$E$$ be a finite set. Let $$d,m,n\in\mathbb N$$. Let $$\mathcal A:=\{A_1,\dots,A_m\}$$ and $$\mathcal B:=\{B_1,\dots,B_n\}$$ be two families of subsets of $$E$$. A partial transversal of $$\mathcal A$$ is the image of an injective function from a subset of $$\{1,\dots,m\}$$ to $$E$$ such that for all $$i$$ in the domain of $$f$$, $$f(i)\in A_i$$. A common partial transversal of $$\mathcal A$$ and $$\mathcal B$$ is a subset that is a partial transversal of $$\mathcal A$$ and $$\mathcal B$$.

The family of partial transversals of $$\mathcal A$$ is (the family of independent sets of) a matroid on $$E$$. (See, for example, Theorem 6.5.2 of Mirsky's book, Transversal Theory.)

If one wants to determine the cardinality of the set of maximum size that is a union of $$d$$ common partial transversals, is there a way of doing so by considering intersections of independent sets arising from possibly different matroid structures on possibly different sets?

The inspiration behind this is that if one has a matroid on a set and one wants to find the subset of maximum size that is a union of $$d$$ independent sets, one can translate this into the problem of finding the subset of maximum size that is independent in two different matroid structures on a different ground set. (See, for example, the end of $$\S6$$ of Chapter 8 of Lawler's book, Combinatorial Optimization: Networks and Matroids.)