On Helly numbers of exponential lattices

Abstract

Given a set S⊆R2, define the Helly number ofS, 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 logα(β) is rational.