  We study string gas dynamics in the early universe and seek to realize the
Brandenberger - Vafa mechanism - a goal that has eluded earlier works - that
singles out three or fewer spatial dimensions as the number which grow large
cosmologically. Considering wound string interactions in an impact parameter
picture, we show that a strong exponential suppression in the interaction rates
for d > 3 spatial dimensions reflects the classical argument that string
worldsheets generically intersect in at most four spacetime dimensions. This
description is appropriate in the early universe if wound strings are heavy -
wrapping long cycles - and diluted. We consider the dynamics of a string gas
coupled to dilaton-gravity and find that a) for any number of dimensions the
universe generically stays trapped in the Hagedorn regime and b) if the
universe fluctuates to a radiation regime any residual winding modes are
diluted enough so that they freeze-out in d > 3 large dimensions while they
generically annihilate for d = 3. In this sense the Brandenberger-Vafa
mechanism is operative.
