On Helly Numbers of Exponential Lattices

Ambrus, Gergely; Balko, Martin; Frankl, Nora; Jung, Attila and Naszodi, Marton (2023). On Helly Numbers of Exponential Lattices. In: Symposium on Computational Geometry (SoCG 2023), 12-15 Jun 2023, Dallas, Texas, USA.

Abstract

Given a set $S \subseteq \mathbb{R}^2$, define the \emph{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 $\mathcal{F}$ of convex sets in~$\mathbb{R}^2$ such that the intersection of any $N$ or fewer members of~$\mathcal{F}$ contains at least one point of $S$, there is a point of $S$ common to all members of $\mathcal{F}$.

We prove that the Helly numbers of \emph{exponential lattices} $\{\alpha^n \colon n \in \mathbb{N}_0\}^2$ are finite for every $\alpha>1$ and we determine their exact values in some instances.
In particular, we obtain $H(\{2^n \colon n \in \mathbb{N}_0\}^2)=5$, solving a problem posed by Dillon (2021).

For real numbers $\alpha, \beta > 1$, we also fully characterize exponential lattices $L(\alpha,\beta) = \{\alpha^n \colon n \in \mathbb{N}_0\} \times \{\beta^n \colon n \in \mathbb{N}_0\}$ with finite Helly numbers by showing that $H(L(\alpha,\beta))$ is finite if and only if $\log_\alpha(\beta)$ is rational.

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