Local equivalence of transversals in matroids

Given any system of n subsets in a matroid M , a transversal of this system is an n-tuple of elements of M , one from each set, which is independent. Two transversals differing by exactly one element are adjacent, and two transversals connected by a sequence of adjacencies are locally equivalent, the distance between them being the minimum number of adjacencies in such a sequence. We give two sufficient conditions for all transversals of a set system to be locally equivalent. Also we propose a conjecture that the distance between any two locally equivalent transversals can be bounded by a function of n only, and provide an example showing that such function, if it exists, must grow at least exponentially. Let M be a matroid, and V = (V1, . . . , Vn) a collection of subsets of M . By a transversal of V we mean a sequence (v1, . . . , vn) of elements of M such that vi ∈ Vi for i = 1, . . . , n, and v1, . . . , vn are independent. By the well-known Rado’s Theorem, transversals exist if and only if the following condition is satisfied: For every X ⊆ {1, . . . , n}, rank ( ⋃