It was known that f (m) exists and is at most exponential.
An advance was made
in 2010 by
Werner and Lenz,
who established a quadratic upper bound, O(m2),
They also uncovered a long history of the problem under
other names, e.g., "d-separated interval piercing."
In fact, the result was already established by Tardos and Karolyi
earlier. See the cited paper for more details.
But, as pointed out by Pablo Soberón,
apparently an earlier result [Kai97, Thm. 1.4],
This largely solves the problem.