Countably Colorful Hyperplane Transversal (2402.10012v1)

Abstract: Let $\left{ \mathcal{F}{n}\right}{n \in \mathbb{N}}$ be an infinite sequence of families of compact connected sets in $\mathbb{R}^{d}$. An infinite sequence of compact connected sets $\left{ B_{n} \right}{n\in \mathbb{N}}$ is called heterochromatic sequence from $\left{ \mathcal{F}{n}\right}{n \in \mathbb{N}}$ if there exists an infinite sequence $\left{ i{n} \right}{n\in \mathbb{N}}$ of natural numbers satisfying the following two properties: (a) ${i{n}}{n\in \mathbb{N}}$ is a monotonically increasing sequence, and (b) for all $n \in \mathbb{N}$, we have $B{n} \in \mathcal{F}{i_n}$. We show that if every heterochromatic sequence from $\left{ \mathcal{F}{n}\right}{n \in \mathbb{N}}$ contains $d+1$ sets that can be pierced by a single hyperplane then there exists a finite collection $\mathcal{H}$ of hyperplanes from $\mathbb{R}^{d}$ that pierces all but finitely many families from $\left{ \mathcal{F}{n}\right}{n \in \mathbb{N}}$. As a direct consequence of our result, we get that if every countable subcollection from an infinite family $\mathcal{F}$ of compact connected sets in $\mathbb{R}^{d}$ contains $d+1$ sets that can be pierced by a single hyperplane then $\mathcal{F}$ can be pierced by finitely many hyperplanes. To establish the optimality of our result we show that, for all $d \in \mathbb{N}$, there exists an infinite sequence $\left{ \mathcal{F}{n}\right}{n \in \mathbb{N}}$ of families of compact connected sets satisfying the following two conditions: (1) for all $n \in \mathbb{N}$, $\mathcal{F}{n}$ is not pierceable by finitely many hyperplanes, and (2) for any $m \in \mathbb{N}$ and every sequence $\left{B_n\right}_{n=m}^{\infty}$ of compact connected sets in $\mathbb{R}^d$, where $B_i\in\mathcal{F}_i$ for all $i \geq m$, there exists a hyperplane in $\mathbb{R}^d$ that pierces at least $d+1$ sets in the sequence.