Colorful Helly Theorem for Piercing Boxes with Two Points

Let $h(d,2)$ denote the smallest integer such that any finite collection of axis parallel boxes in $\mathbb{R}^d$ is two-pierceable if and only if every $h(d,2)$ many boxes from the collection is two-pierceable. Danzer and Gr\"{u}nbaum (1982) proved that $h(d,2)$ equals $3d$ for odd $d$ and $(3d-1)$ for even $d$. In this short note paper, we have given an optimal colorful generalization of the above result, and using it derived a new fractional Helly Theorem for two-piercing boxes in $\mathbb{R}^{d}$. We have also shown that using our fractional Helly theorem and following the same techniques used by Chakraborty et al. (2018) we can design a constant query algorithm for testing if a set of points is $(2,G)$-clusterable, where $G$ is an axis parallel box in $\mathbb{R}^d$.