In the now-inactive (and very nice) "Tricki" website that describes various proof techniques, there is an article (written I think by Timothy Gowers) that describes the following Ramsey-type problem:

Given n axis-aligned rectangles in the plane, what is the largest m such that there are always either m disjoint rectangles, or m rectangles intersecting in one point. The given proof shows that $c \cdot \sqrt{n} / \log(n) \leq m \leq \sqrt{n}$.

http://www.tricki.org/article/Extra_logarithmic_factors

It is mentioned that closing this gap is an open question. Where can this problem (and related background) be found in a research paper or book? Is it still open in this form or was there some progress on it?