Piercing boxes and products of trees

We give a new upper bound on the piercing number $τ(\mathcal{F})$ of a finite family $\mathcal{F}$ of axis-parallel rectangles in the plane when every $p$ members contain an intersecting pair. Namely, we prove $τ(\mathcal{F}) = O(p(\log\log p)(\log\log\log p))$, improving the factor $(\log\log p)^2$