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An (ℵ0, k + 2)-theorem for k-transversals

Chaya Keller,M. Perles

Published 2023 in Israel Journal of Mathematics

ABSTRACT

A family F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{F}$$\end{document} of sets satisfies the (p, q)-property if among every p members of F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{F}$$\end{document}, 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{F}$$\end{document} of compact convex sets in ℝd 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{F}$$\end{document} be an infinite family of closed balls in ℝd, and let 0 ≤ k < d. If among every ℵ0 elements of F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{F}$$\end{document}, some k + 2 can be pierced by a k-dimensional flat, then F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{F}$$\end{document} 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.

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