d-Galvin Families

  • Johan Håstad
  • Guillaume Lagarde
  • Joseph Swernofsky


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 $d$.

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