New Helly-type results for discrete boxes: Quantitative colorful and $(p,q)$-variants

Abstract: In 2008, Halman showed that for any finite set $P\subset \mathbb R^d$ and any finite family $\mathcal{B}$ of axis-parallel boxes in $\mathbb{R}^d$, if the intersection of $P$ and any subfamily $\mathcal{B}' \subseteq\mathcal{B}$ of size at most $2d$ is non-empty, then the intersection of $P$ and $\mathcal{B}$ is also non-empty. Very recently Edwards and Sober\'on initiated the study of quantitative colorful version for $2d$ families, $(p,q)$-type variation for $p\geq q\geq d+1$, and other extensions of this Helly-type result by Halman. In this paper, we study the quantitative colorful Halman problem for $2d-1$ families as well its $(p,q)$-type variation for $p\geq q\geq 2$. Specifically, our main result asserts that for any finite set $P$ and finite families of boxes $\mathcal{B}1,\dots,\mathcal{B}{2d-1}$ in $\mathbb R^d$, where $d\geq 2$, if every transversal $\mathcal{B}$ for the families has an intersection $\bigcap \mathcal{B}$ containing at least $n$ points of $P$, then there exist $j\in[2d-1]$ and a subset of $P$ of size at most [ 2n+\Big\lfloor \frac{n-1}{d \cdot 2^{d-1}} \Big\rfloor, ] such that each box of $\mathcal{B}_j$ contains at least $n$ points of this subset.