  We study the question on whether the famous Golod-Shafarevich estimate, which
gives a lower bound for the Hilbert series of a (noncommutative) algebra, is
attained. This question was considered by Anick in his 1983 paper 'Generic
algebras and CW-complexes', Princeton Univ. Press., where he proved that the
estimate is attained for the number of quadratic relations $d \leq
\frac{n^2}{4}$ and $d \geq \frac{n^2}{2}$, and conjectured that this is the
case for any number of quadratic relations. The particular point where the
number of relations is equal to $ \frac{n(n-1)}{2}$ was addressed by Vershik.
He conjectured that a generic algebra with this number of relations is finite
dimensional.
  We prove that over any infinite field, the Anick conjecture holds for
  $d \geq \frac{4(n^2+n)}{9}$ and arbitrary number of generators $n$, and
confirm the Vershik conjecture over any field of characteristic 0. We give also
a series of related asymptotic results.
