Soumi Nandi: Colorful Helly theorem for piercing boxes with two points

For any natural number $n$, a family $F$ of subsets of a space $X$ is said to be {\em $n$-pierceable}, if there exists $A\subseteq X$ with $|A|\le n$ such that for any $F\in F$, $F\cap A\ne \varnothing$.
Helly's theorem, one of the fundamental results in discrete geometry, says that for any finite family $F$ of convex sets in $R^d$, if every $(d+1)$-tuple from $F$ is $1$-pierceable, then the whole family $F$ is $1$-pierceable. Unfortunately, for $n\ge 2$, a similar statement about the $n$-pierceable sets is not valid for general convex sets. Danzer and Grünbaum proved the first and one of the most important Helly type results on multi-pierceable families; viz. famlies of axis parallel boxes.
One important generalization of Helly's theorem is Colorful Helly's Theorem. In this talk, we shall prove a colorful version of Danzer and Grünbaum's $2$-pierceability result for families of axis parallel boxes. This work was jointly done with Sourav Chakraborty and Arijit Ghosh.