A reminder: this is today.
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On 2022-11-28 10:52, Mykhaylo Tyomkyn wrote:
Dear all,
There will be a noon lecture this Thursday, 1 December, given by Martin Balko. Please find the talk details below.
Best regards, Misha Tyomkyn.
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.
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