Rainbow Matchings and Algebras of Sets

Grinblat (Algebras of Sets and Combinatorics, Translations of Mathematical Monographs, vol. 214. AMS, Providence, 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) = 3 n- 2 for all n≥ 4. In this paper we improve the upper bound (for all large enough n) to v(n) ≤ 16 n/ 5 + O(1).