We give 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：本文给出了 $\mathbb{R}^{d}$ 中无限族坐标轴平行箱体可被有限条坐标轴平行 $k$-平面临戳的充要条件，其中 $0 \leq k < d$。我们还考虑了上述结果的多彩推广并证明了其可行性。本文所研究的问题是 Hadwiger-Debrunner $(p,q)$-问题的无穷变体。