Rainbow matchings and algebras of sets

Grinblat (2002) asks the following question in the context of algebras of sets: What is the smallest number v=v(n) such that, if A₁,..., A_n are n equivalence relations on a common finite ground set X, such that for each i there are at least v elements of X that belong to A_i-equivalence classes of size larger than 1, then X has a rainbow matching-a set of 2n distinct elements a₁, b₁,..., a_n, b_n, such that a_i is A_i-equivalent to b_i for each i? Grinblat has shown that v(n)≤10n/3+O(√n). He asks whether v(n)=3n-2 for all n≥. 4. In this paper we improve the upper bound (for all large enough n) to v(n)≤16n/5+O(1).