# On Helly numbers of exponential lattices

## Martin Balko

*Charles University*

December 1, 2022, 12:20 in S6

## Abstract

For every set $S \subseteq \mathbb{R}^2$, the \emph{Helly number of $S$} is the smallest positive integer $N$, if it exists, such that, for every finite family $\mathcal{F}$ of convex sets in~$\mathbb{R}^2$ the following statement is true: if the intersection of any $N$ or fewer members of~$\mathcal{F}$ contains at least one point of $S$, then $\bigcap \mathcal{F}$ contains at least one point of $S$.

We prove that the Helly numbers of 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, which in particular solves a problem posed by Dillon. We also fully characterize nondiagonal exponential lattices $\{\alpha^n \colon n \in \mathbb{N}_0\} \times \{\beta^n \colon n \in \mathbb{N}_0\}$ with $\alpha,\beta>1$ with finite Helly numbers using results from number theory.

This is a joint work with Gergely Ambrus, Nóra Frankl, Attila Jung, and Márton Naszódi.