## Abstract

Given a set S⊆R^{2}, define the 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 F of convex sets in R^{2} 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 {α^{n}:n∈N_{0}}^{2} are finite for every α>1 and we determine their exact values in some instances. In particular, we obtain H({2^{n}:n∈N_{0}}^{2})=5, solving a problem posed by Dillon (2021). For real numbers α,β>1, we also fully characterize exponential lattices L(α,β)={α^{n}:n∈N_{0}}×{β^{n}:n∈N_{0}} with finite Helly numbers by showing that H(L(α,β)) is finite if and only if log_{α}(β) is rational.

Original language | English |
---|---|

Article number | 103884 |

Journal | European Journal of Combinatorics |

Volume | 116 |

DOIs | |

State | Published - 1 Feb 2024 |

Externally published | Yes |

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics