  It follows from the Golod--Shafarevich theorem that if R is an associative
algebra given by n generators and $d<\frac{n^2}{4}\cos^{-2}(\frac{\pi}{k+1})$
quadratic relations, then R is not k-step nilpotent. We show that the above
estimate is asymptotically optimal, and establish number of related results.
For example, we show that for any k this estimate is attained for ifinitely
many n.
