An (ℵ_0,k+2)-Theorem for k-Transversals
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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 (ℵ_0,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 ℵ_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. The same result holds also for a wider class of families which consist of 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.
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