  Motivated by a question of Grinblat, we study the minimal number
$\mathfrak{v}(n)$ that satisfies the following. If $A_1,\ldots, A_n$ are
equivalence relations on a set $X$ such that for every $i\in[n]$ there are at
least $\mathfrak{v}(n)$ elements whose equivalence classes with respect to
$A_i$ are nontrivial, then $A_1, \ldots, A_n$ contain a rainbow matching, i.e.
there exist $2n$ distinct elements $x_1,y_1,\ldots,x_n,y_n\in X$ with
$x_i\sim_{A_i} y_i$ for each $i\in [n]$. Grinblat asked whether
$\mathfrak{v}(n) = 3n-2$ for every $n\geq 4$. The best-known upper bound was
$\mathfrak{v}(n) \leq 16n/5 + \mathcal{O}(1)$ due to Nivash and Omri.
Transferring the problem into the setting of edge-coloured multigraphs, we
affirm Grinblat's question asymptotically, i.e. we show that $\mathfrak{v}(n) =
3n+o(n)$.
