  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.
