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, . . . , An 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 Ai-equivalence classes of size larger than 1, then X has a rainbow matching—a set of 2n distinct elements a1, b1, . . . , an, bn, such that ai is Ai-equivalent to bi 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).