$d$-Galvin families

Abstract

The Galvin problem asks for the minimum size of a family $\mathcal{F}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{n/2}\right)$ 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 $\mathcal{F}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{n/d}\right)$ 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 $d$.

9 pages, 6 figures