Rainbow matchings and algebras of sets

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 $A_1,\dots ,A_n$ 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 $a_1, b_1, \ldots, a_n, b_n$,suchthat$a_i$is$A_i$-equivalentto$b_i$foreach$i$? Grinblat has shown that $\mathfrak{v}(n)\le 10n\mathrm{/}3+O(\sqrt{n})$. 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 $\mathfrak{v}(n)\le 16n\mathrm{/}5+O(1)$.