  The Galvin problem asks for the minimum size of a family $\mathcal{F}
\subseteq \binom{[n]}{n/2}$ with the property that, for any set $A$ of size
$\frac n 2$, there is a set $S \in \mathcal{F}$ which is balanced on $A$,
meaning that $|S \cap A| = |S \cap \overline{A}|$. We consider a generalization
of this question that comes from a possible approach in complexity theory. In
the generalization the required property is, for any $A$, to be able to find
$d$ sets from a family $\mathcal{F} \subseteq \binom{[n]}{n/d}$ that form a
partition of $[n]$ and such that each part is balanced on $A$. We construct
such families of size polynomial in the parameters $n$ and $d$.
