d-Galvin Families.

The Galvin problem asks for the minimum size of a family F subset of ([n]n/2) with the property that, for any set A of size n/2, there is a set S is an element of F which is balanced on A, meaning that vertical bar S boolean AND A vertical bar = vertical bar S boolean AND ( A) over bar vertical bar. 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 F subset of ([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.