An (ℵ₀, 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 (ℵ₀, k + 2)-theorem with respect to k-transversals: Let F be an infinite family of closed balls in ℝ^d, and let 0 ≤ k < d. If among every ℵ₀ 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. We derive this result as a corollary of a more general result which proves the same assertion for families of not necessarily convex objects called near-balls, to be defined below. This is the first (p, q)-theorem in which the assumption is weakened to an (∞, ·) assumption. Our proofs combine geometric and topological tools.