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  LeibnizZentrum für Informatik; Dagstuhl Publishing, Leibniz, No.8.

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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 logα(β)logα(β) is rational.
Item Type:  Book Section 

Subjects:  Q Science / természettudomány > QA Mathematics / matematika Q Science / természettudomány > QA Mathematics / matematika > QA73 Geometry / geometria 
SWORD Depositor:  MTMT SWORD 
Depositing User:  MTMT SWORD 
Date Deposited:  30 Mar 2023 13:15 
Last Modified:  27 Sep 2023 12:26 
URI:  http://real.mtak.hu/id/eprint/163158 
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