An (?₀, k + 2)-Theorem for k-Transversals

A family F of sets satisfies the (p, q)-property if among every p members of F, some q can be pierced by a single point. The celebrated (p, q)-theorem of Alon and Kleitman asserts that for any p ? q ? d + 1, any family F of compact convex sets in R^d that satisfies the (p, q)-property can be pierced by a finite number c(p, q, d) of points. A similar theorem with respect to piercing by (d - 1)-dimensional flats, called (d - 1)-transversals, was obtained by Alon and Kalai. In this paper we prove the following result, which can be viewed as an (?0, k + 2)-theorem with respect to k-transversals: Let F be an infinite family of sets in R^d such that each A ? F contains a ball of radius r and is contained in a ball of radius R, and let 0 ? k < d. If among every ?0 elements of F, some k + 2 can be pierced by a k-dimensional flat, then F can be pierced by a finite number of k-dimensional flats. This is the first (p, q)-theorem in which the assumption is weakened to an (8, ·) assumption. Our proofs combine geometric and topological tools.