On Helly Numbers of Exponential Lattices

Ambrus, Gergely and Balko, Martin and Frankl, Nóra and Jung, Attila and Naszódi, Márton (2023) On Helly Numbers of Exponential Lattices. In: 39th International Symposium on Computational Geometry (SoCG 2023). Schloss Dagstuhl - Leibniz-Zentrum für Informatik; Dagstuhl Publishing, Leibniz, No.-8.

Preview	Text LIPIcs-SoCG-2023-8.pdf - Published Version Download (900kB) | Preview
Preview	Text 2301.04683v2.pdf - Accepted Version Download (666kB) | Preview

Abstract

Given a set S⊆R2S⊆R2, define the Helly number of SS, denoted by H(S)H(S), as the smallest positive integer NN, if it exists, for which the following statement is true: For any finite family FF of convex sets in~R2R2 such that the intersection of any NN or fewer members of~FF contains at least one point of SS, there is a point of SS common to all members of FF. We prove that the Helly numbers of exponential lattices {αn ⁣:n∈N0}2{αn:n∈N0​}2 are finite for every α>1α>1 and we determine their exact values in some instances. In particular, we obtain H({2n ⁣:n∈N0}2)=5H({2n:n∈N0​}2)=5, solving a problem posed by Dillon (2021). For real numbers α,β>1α,β>1, we also fully characterize exponential lattices L(α,β)={αn ⁣:n∈N0}×{βn ⁣:n∈N0}L(α,β)={αn:n∈N0​}×{βn:n∈N0​} with finite Helly numbers by showing that H(L(α,β))H(L(α,β)) is finite if and only if log⁡α(β)logα​(β) is rational.

Actions (login required)

Edit Item