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 A_1, … , 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 2 n distinct elements a_1, b_1, … , 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) .