Document Open Access Logo

On Helly Numbers of Exponential Lattices

Authors Gergely Ambrus, Martin Balko, Nóra Frankl, Attila Jung, Márton Naszódi

Thumbnail PDF


  • Filesize: 0.85 MB
  • 16 pages

Document Identifiers

Author Details

Gergely Ambrus
  • Department of Geometry, Bolyai Institute, University of Szeged, Hungary
  • Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Martin Balko
  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Nóra Frankl
  • School of Mathematics and Statistics, The Open University, Milton Keynes, UK
  • Alfréd Rényi Institute of Mathematics, Budapest, Hungary
Attila Jung
  • Institute of Mathematics, ELTE Eötvös Loránd University, Budapest, Hungary
Márton Naszódi
  • Department of Geometry, ELTE Eötvös Loránd University, Budapest, Hungary
  • MTA-ELTE Lendület Combinatorial Geometry Research Group, Budapest, Hungary


This research was initiated at the 11th Emléktábla workshop on combinatorics and geometry. We would like to thank Géza Tóth for interesting discussions about the problem during the early stages of the research

Cite AsGet BibTex

Gergely Ambrus, Martin Balko, Nóra Frankl, Attila Jung, and Márton Naszódi. On Helly Numbers of Exponential Lattices. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 8:1-8:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


Given a set S ⊆ ℝ², define the Helly number of S, denoted by H(S), as the smallest positive integer N, if it exists, for which the following statement is true: for any finite family ℱ of convex sets in ℝ² such that the intersection of any N or fewer members of ℱ contains at least one point of S, there is a point of S common to all members of ℱ. We prove that the Helly numbers of exponential lattices {αⁿ : n ∈ ℕ₀}² are finite for every α > 1 and we determine their exact values in some instances. In particular, we obtain H({2ⁿ : n ∈ ℕ₀}²) = 5, solving a problem posed by Dillon (2021). For real numbers α, β > 1, we also fully characterize exponential lattices L(α,β) = {αⁿ : n ∈ ℕ₀} × {βⁿ : n ∈ ℕ₀} with finite Helly numbers by showing that H(L(α,β)) is finite if and only if log_α(β) is rational.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Helly numbers
  • exponential lattices
  • Diophantine approximation


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


  1. Nina Amenta, Jesús A. De Loera, and Pablo Soberón. Helly’s theorem: new variations and applications. In Algebraic and geometric methods in discrete mathematics, volume 685 of Contemp. Math., pages 55-95. Amer. Math. Soc., Providence, RI, 2017. URL:
  2. Gennadiy Averkov, Bernardo González Merino, Ingo Paschke, Matthias Schymura, and Stefan Weltge. Tight bounds on discrete quantitative Helly numbers. Adv. in Appl. Math., 89:76-101, 2017. URL:
  3. David E. Bell. A theorem concerning the integer lattice. Studies in Appl. Math., 56(2):187-188, 1976/77. URL:
  4. Michele Conforti and Marco Di Summa. Maximal S-free convex sets and the Helly number. SIAM J. Discrete Math., 30(4):2206-2216, 2016. URL:
  5. Jesús A. De Loera, Reuben N. La Haye, Déborah Oliveros, and Edgardo Roldán-Pensado. Helly numbers of algebraic subsets of ℝ^d and an extension of Doignon’s theorem. Adv. Geom., 17(4):473-482, 2017. URL:
  6. Jesús A. De Loera, Reuben N. La Haye, David Rolnick, and Pablo Soberón. Quantitative Tverberg theorems over lattices and other discrete sets. Discrete Comput. Geom., 58(2):435-448, 2017. URL:
  7. Travis Dillon. Discrete quantitative Helly-type theorems with boxes. Adv. in Appl. Math., 129:Paper No. 102217, 17, 2021. URL:
  8. Jean-Paul Doignon. Convexity in cristallographical lattices. J. Geom., 3:71-85, 1973. URL:
  9. Alexey Garber. On Helly number for crystals and cut-and-project sets. Arxiv preprint, 2017.
  10. Jaroslav Hančl and Ondřej Turek. One-sided Diophantine approximations. Journal of Physics A: Mathematical and Theoretical, 52(4):045205, January 2019. URL:
  11. Eduard Helly. Über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jahresber. Deutsch. Math.-Verein., 32:175-176, 1923. Google Scholar
  12. Alan J. Hoffman. Binding constraints and Helly numbers. In Second International Conference on Combinatorial Mathematics (New York, 1978), volume 319 of Ann. New York Acad. Sci., pages 284-288. New York Acad. Sci., New York, 1979. Google Scholar
  13. Andreas Holmsen and Rephael Wenger. Helly-type theorems and geometric transversals. In Handbook of Discrete and Computational Geometry (3rd ed.). CRC Press, 2017. Google Scholar
  14. Aleksandr Ya. Khinchin. Continued fractions. Dover Publications, Inc., Mineola, NY, Russian edition, 1997. With a preface by B. V. Gnedenko, reprint of the 1964 translation. Google Scholar
  15. Herbert E. Scarf. An observation on the structure of production sets with indivisibilities. Proc. Nat. Acad. Sci. U.S.A., 74(9):3637-3641, 1977. URL:
  16. Kevin Barrett Summers. The Helly Number of the Prime-coordinate Point Set. Bachelor’s thesis, University of California, 2015. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail