Fair representation in the intersection of two matroids

Mysteriously, hypergraphs that are the intersection of two matroids behave in some respects almost as well as one matroid. In the present paper we study one such phenomenon - the surprising ability of the intersection of two matroids to fairly represent the parts of a given partition of the ground set. For a simplicial complex C denote by beta(C) the minimal number of edges from C needed to cover the ground set. If C is a matroid then for every partition A(1) , . . . , A(m) of the ground set there exists a set S is an element of C meeting each A(i) in at least left perpendicular vertical bar A(i)vertical bar/beta(C)right perpendicular elements. We conjecture that a slightly weaker result is true for the intersection of two matroids: if D = P boolean AND Q, where P, Q are matroids on the same ground set V and beta(P), beta(Q) <= k, then for every partition A(1) , . . . , A(m) of the ground set there exists a set S is an element of D meeting each A(i) in at least 1/k vertical bar A(i)vertical bar - 1 elements. We prove that if m = 2 (meaning that the partition is into two sets) there is a set belonging to D meeting each A(i) in at least (1/k -1/vertical bar V vertical bar) vertical bar A(i)vertical bar - 1 elements.