  Shelah Spencer [ShSp:304] proved the 0-1 law for the random graphs G(n,p_n),
p_n=n^{- alpha}, alpha in (0,1) irrational (set of nodes in [n]= {1, ...,n},
the edges are drawn independently, probability of edge is p_n). One may wonder
what can we say on sentences psi for which Prob (G(n,p_n) models psi) converge
to zero, Lynch asked the question and did the analysis, getting (for every
psi):
  EITHER [(alpha)] Prob [G(n,p_n) models psi]=cn^{- beta} + O(n^{-
beta-epsilon}) for some epsilon such that beta >epsilon>0
  OR [(beta)] Prob (G(n,p_n) models psi)= O(n^{-epsilon}) for every epsilon>0.
  Lynch conjectured that in case (beta) we have
  [(beta^+)] Prob (G(n,p_n) models psi)= O(e^{-n^epsilon}) for some epsilon>0.
We prove it here.
