A note on piercing discrete rectangles

In 2008, Halman proved a discrete Helly-type theorem for axis-parallel boxes in $\mathbb R^d$. Very recently, this result was extended to the $(p,q)$ setting with $p \geq q \geq d+1$ by Edwards and Sober\'on, and subsequently to the case $p \geq q \geq 2$ by Gangopadhyay, Polyanskii, and the author of this paper. In this paper, we obtain improved bounds for the $(p,q)$ problem in the case $q=2$ and $d=2$. More precisely, our main result asserts that for any integer $p \geq 2$, any set $P \subseteq \mathbb R^2$, and any finite family $\mathcal B$ of axis-parallel rectangles in $\mathbb R^2$ such that every rectangle contains a point of $P$, if among every $p$ rectangles there exist two whose intersection contains a point of $P$, then there exists a subset $S \subseteq P$ of size at most $O\!\bigl( (p \log \log p)^2 \bigr)$ such that every rectangle contains a point of $S$. Moreover, when $p=2$, the size of $S$ can be bounded by $8$.