  In 1973 Erdos asked whether there are n-vertex partial Steiner triple systems
with arbitrary high girth and quadratically many triples. (Here girth is
defined as the smallest integer g \ge 4 for which some g-element vertex-set
contains at least g-2 triples.)
  We answer this question, by showing existence of approximate Steiner triple
systems with arbitrary high girth. More concretely, for any fixed \ell \ge 4 we
show that a natural constrained random process typically produces a partial
Steiner triple system with (1/6-o(1))n^2 triples and girth larger than \ell.
The process iteratively adds random triples subject to the constraint that the
girth remains larger than \ell. Our result is best possible up to the
o(1)-term, which is a negative power of n.
