In this short note, we provide the necessary and sufficient condition for an infinite collection of axis-parallel boxes in $\mathbb{R}^{d}$ to be pierceable by finitely many axis-parallel $k$-flats, where $0 \leq k < d$. We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner $(p, q)$-problem.

翻译：在这篇短文中，我们给出了$\mathbb{R}^{d}$中无限个轴平行盒子能被有限多个轴平行$k$-平面刺穿的充分必要条件，其中$0 \leq k < d$。我们还考虑了上述结果的彩色推广形式，并证明了其可行性。本文所研究的问题是Hadwiger-Debrunner $(p, q)$-问题的一个无限变体。