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On Helly Numbers of Exponential Lattices

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

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

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Abstract

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.

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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) https://doi.org/10.4230/LIPIcs.SoCG.2023.8

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

Funding

  • Ambrus, Gergely: Partially supported by ERC Advanced Grant "GeoScape", by the Hungarian National Research grant no. NKFIH KKP-133819, and by project no. TKP2021-NVA-09, which has been implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme.
  • Balko, Martin: Supported by the grant no. 21/32817S of the Czech Science Foundation (GAČR) and by the Center for Foundations of Modern Computer Science (Charles University project UNCE/SCI/004). This article is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 810115).
  • Frankl, Nóra: Partially supported by ERC Advanced Grant "GeoScape".
  • Jung, Attila: Supported by the Rényi Doctoral Fellowship of the Rényi Institute.
  • Naszódi, Márton: Supported by the J{á}nos Bolyai Scholarship of the Hungarian Acadamy of Sciences.

Acknowledgements

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

References

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