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 . 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).