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An (ℵ₀,k+2)-Theorem for k-Transversals

Authors Chaya Keller , Micha A. Perles

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Author Details

Chaya Keller
  • Ariel University, Israel
Micha A. Perles
  • Einstein Institute of Mathematics, Hebrew University, Jerusalem, Israel

Funding

  • Keller, Chaya: Research partially supported by the Israel Science Foundation (grant no. 1065/20).

Acknowledgements

The authors are grateful to Andreas Holmsen for valuable suggestions and information.

Cite AsGet BibTex

Chaya Keller and Micha A. Perles. An (ℵ₀,k+2)-Theorem for k-Transversals. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 50:1-50:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.50

Abstract

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 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 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. This is the first (p,q)-theorem in which the assumption is weakened to an (∞,⋅) assumption. Our proofs combine geometric and topological tools.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • convexity
  • (p,q)-theorem
  • k-transversal
  • infinite (p,q)-theorem

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