Document Open Access Logo

A Stepping-Up Lemma for Topological Set Systems

Authors Xavier Goaoc, Andreas F. Holmsen, Zuzana Patáková

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2021.40.pdf
  • Filesize: 0.73 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Helly-type theorem
  • Topological combinatorics
  • Homological minors
  • Stair convexity
  • Cubical complexes
  • Homological VC dimension
  • Ramsey-type theorem

Metrics

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

Abstract

Intersection patterns of convex sets in ℝ^d have the remarkable property that for d+1 ≤ k ≤ 𝓁, in any sufficiently large family of convex sets in ℝ^d, if a constant fraction of the k-element subfamilies have nonempty intersection, then a constant fraction of the 𝓁-element subfamilies must also have nonempty intersection. Here, we prove that a similar phenomenon holds for any topological set system ℱ in ℝ^d. Quantitatively, our bounds depend on how complicated the intersection of 𝓁 elements of ℱ can be, as measured by the maximum of the ⌈d/2⌉ first Betti numbers. As an application, we improve the fractional Helly number of set systems with bounded topological complexity due to the third author, from a Ramsey number down to d+1. We also shed some light on a conjecture of Kalai and Meshulam on intersection patterns of sets with bounded homological VC dimension. A key ingredient in our proof is the use of the stair convexity of Bukh, Matoušek and Nivasch to recast a simplicial complex as a homological minor of a cubical complex.

Cite As Get BibTex

Xavier Goaoc, Andreas F. Holmsen, and Zuzana Patáková. A Stepping-Up Lemma for Topological Set Systems. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.SoCG.2021.40

Author Details

Xavier Goaoc
  • LORIA, Université de Lorraine, France
Andreas F. Holmsen
  • Department of Mathematical Sciences, KAIST, Daejeon, South Korea
Zuzana Patáková
  • Department of Algebra, Faculty of Mathematics and Physics, Charles University, Czech Republic

Funding

  • Goaoc, Xavier: Institut Universitaire de France.
  • Holmsen, Andreas F.: NRF grant No. 2020R1F1A1A0104849011.
  • Patáková, Zuzana: Charles University projects PRIMUS/21/SCI/014 and UNCE/SCI/022.

References

  1. N. Alon and G. Kalai. A simple proof of the upper bound theorem. European Journal of Combinatorics, 6(3):211-214, 1985. Google Scholar
  2. N. Alon, G. Kalai, J. Matoušek, and R. Meshulam. Transversal numbers for hypergraphs arising in geometry. Adv. in Appl. Math., 29(1):79-101, 2002. Google Scholar
  3. N. Amenta. Helly theorems and generalized linear programming. Discrete Comput. Geom., 12:241 - -261, 1994. Google Scholar
  4. I. Bárány. A generalization of Carathéodory’s theorem. Discrete Math., 40(2-3):141-152, 1982. Google Scholar
  5. I. Bárány and J. Matoušek. A fractional Helly theorem for convex lattice sets. Adv. Math., 174(2):227-235, 2003. Google Scholar
  6. B. Bukh, J. Matoušek, and G. Nivasch. Lower bounds for weak epsilon-nets and stair-convexity. Israel J. Math., 182:199-208, 2011. Google Scholar
  7. S. Chakraborty, R. Pratap, S. Roy, and S. Saraf. Helly-type theorems in property testing. International Journal of Computational Geometry & Applications, 28(04):365-379, 2018. Google Scholar
  8. L. Danzer, B. Grünbaum, and V. Klee. Helly’s theorem and its relatives. In Proc. Sympos. Pure Math., Vol. VII, pages 101-180. Amer. Math. Soc., Providence, R.I., 1963. Google Scholar
  9. J. De Loera, X. Goaoc, F. Meunier, and N. Mustafa. The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg. Bulletin of the American Mathematical Society, 56(3):415-511, 2019. Google Scholar
  10. É. C. De Verdière, G. Ginot, and X. Goaoc. Helly numbers of acyclic families. Adv. Math., 253:163-193, 2014. Google Scholar
  11. J. Eckhoff. An upper-bound theorem for families of convex sets. Geometriae Dedicata, 19(2):217-227, 1985. Google Scholar
  12. J. Eckhoff. Helly, Radon, and Carathéodory type theorems. In Handbook of convex geometry, Vol. A, B, pages 389-448. North-Holland, Amsterdam, 1993. Google Scholar
  13. P. Erdős and M. Simonovits. Supersaturated graphs and hypergraphs. Combinatorica, 3(2):181-192, 1983. URL: https://doi.org/10.1007/BF02579292.
  14. X. Goaoc, P. Paták, Z. Patáková, M. Tancer, and U. Wagner. Bounding Helly numbers via Betti numbers. In A journey through discrete mathematics, pages 407-447. Springer, Cham, 2017. Google Scholar
  15. R. L. Graham, B. L. Rothschild, and J. H. Spencer. Ramsey theory, volume 20. John Wiley & Sons, 1990. Google Scholar
  16. S. Hell. Tverberg-type theorems and the fractional Helly property, 2006. PhD thesis. Google Scholar
  17. A. Holmsen and D. Lee. Radon numbers and the fractional Helly theorem, 2019. URL: http://arxiv.org/abs/1903.01068.
  18. A. F. Holmsen, M. Kim, and S. Lee. Nerves, minors, and piercing numbers. Transactions of the American Mathematical Society, 371(12):8755-8779, 2019. Google Scholar
  19. T. Kaczynski, K. Mischaikow, and M. Mrozek. Computational homology, volume 157. Springer Science & Business Media, 2006. Google Scholar
  20. G. Kalai. Intersection patterns of convex sets. Israel Journal of Mathematics, 48(2-3):161-174, 1984. Google Scholar
  21. G. Kalai. Combinatorial expectations from commutative algebra. In I. Peeva and V. Welker, editors, Combinatorial Commutative Algebra, volume 1(3), pages 1729-1734. Oberwolfach Reports, 2004. Google Scholar
  22. G. Kalai. Problems for Imre Bárány’s birthday, 2017. URL: https://gilkalai.wordpress.com/2017/05/23/problems-for-imre-baranys-birthday/.
  23. G. Kalai and R. Meshulam. A topological colorful Helly theorem. Adv. Math., 191(2):305-311, 2005. Google Scholar
  24. G. Kalai and R. Meshulam. Leray numbers of projections and a topological Helly-type theorem. Journal of Topology, 1(3):551-556, 2008. Google Scholar
  25. G. Kalai and Z. Patáková. Intersection patterns of planar sets. Discrete & Computational Geometry, 64:304-323, 2020. Google Scholar
  26. M. Katchalski and A. Liu. A problem of geometry in ℝⁿ. Proceedings of the American Mathematical Society, 75(2):284-288, 1979. Google Scholar
  27. J. Matoušek. A Helly-type theorem for unions of convex sets. Discrete & Computational Geometry, 18(1):1-12, 1997. Google Scholar
  28. J. Matousek. Lectures on discrete geometry, volume 212. Springer Science & Business Media, 2013. Google Scholar
  29. Z. Patáková. Bounding Radon Number via Betti Numbers. In Sergio Cabello and Danny Z. Chen, editors, 36th International Symposium on Computational Geometry (SoCG 2020), volume 164 of Leibniz International Proceedings in Informatics (LIPIcs), pages 61:1-61:13. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  30. U. Wagner. Minors in random and expanding hypergraphs. In Proceedings of the 27th Annual Symposium on Computational Geometry (SoCG), pages 351-360, 2011. Google Scholar
  31. Gerd Wegner. d-collapsing and nerves of families of convex sets. Archiv der Mathematik, 26(1):317-321, 1975. 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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact