Improved Helly numbers of product sets

TL;DR

The paper investigates Helly numbers of product sets $S=A^d$ in discrete convex geometry, with a focus on exponential lattices $L_d(α)$ and arithmetic-congruence constructions. It introduces a sharp geometric upper bound for $h(L_2(α))$ and a dimension-dependent lower bound, showing $h(L_d(α)) ≥ binom(k+d-1}{d-1}$ with $k=floor(sqrt{1/(α-1)})$, highlighting the richer behavior in higher dimensions. The authors also demonstrate that 2-syndetic sets can yield $A^2$ with an infinite Helly number and prove a modular arithmetic bound: for $A=S+m\mathbb{Z}$ with $|S|=k$ and certain $d$, $h(A^d) ≤ k^d$, along with corollaries for prime moduli and a concrete example. Together, these results advance understanding of finite Helly-type theorems on sparse product sets and raise several open questions about higher dimensions and congruence-based constructions.

Abstract

A finite family $\mathcal F$ of convex sets is $k$-intersecting in $S \subseteq \mathbb{R}^d$ if the intersection of every subset of $k$ convex sets in $\mathcal F$ contains a point in $S$. The Helly number of $S$ is the minimum $k$, if it exists, such that every $k$-intersecting family contains a point of $S$ in its intersection. In this paper, we improve bounds on the Helly number of product sets of the form $A^d$ for various sets $A \subseteq \mathbb{R}$, including the ``exponential grid'' $A = \{α^n : n \in \mathbb{N}\}$ and sets $A\subseteq \mathbb{Z}$ defined by congruence relations.

Figures (2)

• Figure 1: The named points in the proof of Theorem \ref{['expLatStronger']}.
• Figure 2: An $8$-vertex empty polygon in $(\{0,1\}+3\mathbb{Z}\mspace{2mu})^2$.

Theorems & Definitions (18)

• Theorem : Doignon's theorem
• Theorem : Hoffman Hoffman1979
• Theorem : Ambrus, Balko, Frankl, Jung, Naszódi Ambrus
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Lemma 2.1: Lemma 10 and Corollary 11 in Ambrus
• proof : Proof of Theorem \ref{['expLatStronger']}
• Corollary 2.2