  The intersection of two Steiner triple systems (X,A) and (X,B) is the set A
intersect B. The fine intersection problem for Steiner triple systems is to
determine for each v, the set I(v), consisting of all possible pairs (m,n) such
that there exist two Steiner triple systems of order v whose intersection has n
blocks over m points. We show that for v = 1 or 3 (mod 6), |I(v)| = Omega(v^3),
where previous results only imply that |I(v)| = Omega(v^2).
