For which α there are first order graph statements A of given quantifier depth k such that a Zero-One law does not hold for the random graph G ( n , p ( n ) ) with p ( n ) at or near (there are two notions) n - α ? A fairly complete description is given in both the near dense ( α near zero) and near linear ( α near one) cases.
['Joel Spencer', 'M. E. Zhukovskii']
Bounded quantifier depth spectra for random graphs