  Grinblat (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 $\mathfrak v(n) \le 16n/5 + O(1)$.
