Document Open Access Logo

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

Authors Chaya Keller , Micha A. Perles

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2022.50.pdf
  • Filesize: 0.79 MB
  • 14 pages

Document Identifiers

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 As Get 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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. N. Alon and G. Kalai. Bounding the piercing number. Discrete Comput. Geom., 13:245-256, 1995. Google Scholar
  2. N. Alon, G. Kalai, J. Matoušek, and R. Meshulam. Transversal numbers for hypergraphs arising in geometry. Adv. Appl. Math., 29:79-101, 2002. Google Scholar
  3. N. Alon and D. J. Kleitman. Piercing convex sets and the Hadwiger-Debrunner (p,q)-problem. Advances in Mathematics, 96(1):103-112, 1992. Google Scholar
  4. N. Alon, J. Pach, R. Pinchasi, R. Radoic̆ić, and M. Sharir. Crossing patterns of semi-algebraic sets. J. Combin. Theory, Ser. A, 111:310-326, 2005. Google Scholar
  5. J. L. Arocha, J. Bracho, and L. Montejano. Flat transversals to flats and convex sets of a fixed dimension. Advances in Mathematics, 213(2):902-918, 2007. Google Scholar
  6. B. Aronov, J. E. Goodman, and R. Pollack. A Helly-type theorem for higher-dimensional transversals. Comput. Geom., 21:177-183, 2002. Google Scholar
  7. I. Bárány and G. Kalai. Helly-type problems, 2021. URL: http://arxiv.org/abs/2108.08804.
  8. V. Boltyanski and A. Soifer. Geometric études in combinatorial mathematics. Center for Excellence in Mathematical Education, Colorado Springs, CO, 1991. Google Scholar
  9. L. Danzer. Zur lösung des gallaischen problems über kreisscheiben in der euklidischen ebene. Studia Sci. Math. Hungar., 21:111-134, 1986. Google Scholar
  10. L. Danzer, B. Grünbaum, and V. Klee. Helly’s theorem and its relatives. In V. Klee, editor, Convexity, Proceedings of Symposium in Pure Mathematics, volume 7, pages 100-181. American Mathematical Society, Providence, RI, 1963. Google Scholar
  11. A. Dumitrescu and M. Jiang. Piercing translates and homothets of a convex body. Algorithmica, 61(1):94-115, 2011. Google Scholar
  12. B. Dushnik and E. W. Miller. Partially ordered sets. American Journal of Mathematics, 63(3):600-610, 1941. Google Scholar
  13. J. Eckhoff. A survey of the Hadwiger-Debrunner (p, q)-problem. In B. Aronov, S. Basu, J. Pach, and M. Sharir, editors, Discrete and Computational Geometry, volume 25 of Algorithms and Combinatorics, pages 347-377. Springer Berlin Heidelberg, 2003. Google Scholar
  14. P. Erdős and R. Rado. A partition calculus in set theory. Bulletin of the American Mathematical Society, 62(5):427-489, 1956. Google Scholar
  15. J. Fox, J. Pach, and C. D. Tóth. Intersection patterns of curves. J. London Math. Society, 83:389-406, 2011. Google Scholar
  16. S. Gao and S. Zerbib. The (2,2) and (4,3) properties in families of fat sets in the plane. SIAM Journal of Discrete Math., 33(3):1326-1337, 2019. Google Scholar
  17. B. Grünbaum. On intersections of similar sets. Portugaliae Mathematica, 18:155-164, 1959. Google Scholar
  18. H. Hadwiger and H. Debrunner. Über eine variante zum Hellyschen satz. Archiv der Mathematik, 8(4):309-313, 1957. Google Scholar
  19. E. Helly. Uber mengen konvexer körper mit gemeinschaftlichen punkte. Jahresbericht der Deutschen Mathematiker-Vereinigung, 32:175-176, 1923. Google Scholar
  20. A. Holmsen and R. Wenger. Helly-type theorems and geometric transversals. In J. O'Rourke J. E. Goodman and C. D. Tóth, editors, Handbook of Discrete and Computational Geometry, 3rd Edition, pages 91-123. CRC Press LLC, Boca Raton, FL, 2017. Google Scholar
  21. A. F. Holmsen and D. Lee. Radon numbers and the fractional Helly theorem. Isr. J. Math., 241:433-447, 2021. Google Scholar
  22. S. J. Kim, K. Nakprasit, M.J. Pelsmajer, and J. Skokan. Transversal numbers of translates of a convex body. Discrete Math., 306:2166-2173, 2006. Google Scholar
  23. D. Larman, J. Matoušek, J. Pach, and J. Töröcsik. A ramsey-type result for planar convex sets. Bulletin of London Math. Soc., 26:132-136, 1994. Google Scholar
  24. J. Matoušek. Bounded VC-dimension implies a fractional Helly theorem. Discrete Comput. Geom., 31(2):251-255, 2004. Google Scholar
  25. A. Montejano, L. Montejano, E. Roldán-Pensado, and P. Soberón. About an Erdős-Grünbaum conjecture concerning piercing of non-bounded convex sets. Discrete Comput. Geom., 53(4):941-950, 2015. Google Scholar
  26. S. Moran and A. Yehudayoff. On weak ε-nets and the Radon number. Discret. Comput. Geom., 64(4):1125-1140, 2020. Google Scholar
  27. T. Müller. A counterexample to a conjecture of Grünbaum on piercing convex sets in the plane. Discrete Math., 313(24):2868-2871, 2013. Google Scholar
  28. Z. Patáková. Bounding Radon number via Betti numbers. In 36th International Symposium on Computational Geometry, SoCG 2020, volume 164 of LIPIcs, pages 61:1-61:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  29. R. Pinchasi. A note on smaller fractional Helly numbers. Discrete Comput. Geom., 54:663-668, 2015. Google Scholar
  30. F. P. Ramsey. On a problem of formal logic. Proc. London Math. Soc. (Ser. 2), 30(1):264-286, 1930. Google Scholar
  31. L. Santaló. Un teorema sobre conjuntos de paralelepipedos de aristas paralelas. Publ. Inst. Mat. Univ. Nac. Litoral, 2:49-60, 1940. Google Scholar
  32. P. Vincensini. Figures convexes et variétés linéaires de l’espace euclidien à n dimensions. Bull Sci. Math., 59:163-174, 1935. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact