Rainbow Matchings and Algebras of Sets

Abstract

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 $\mathfrak {v} = \mathfrak {v}(n)$ such that, if $A_1, \ldots , A_n$ are n equivalence relations on a common finite ground set X, such that for each i there are at least $\mathfrak {v}$ 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 $a_1, b_1, \ldots , a_n, b_n$, such that $a_i$ is $A_i$-equivalent to $b_i$ for each i? Grinblat has shown that $\mathfrak {v}(n) \le 10n/3 + O(\sqrt{n})$. He asks whether $\mathfrak {v}(n) = 3n-2$ for all $n\ge 4$. In this paper we improve the upper bound (for all large enough n) to $\mathfrak {v}(n) \le 16n/5 + O(1)$.