Zero-one laws for existential first order sentences of bounded quantifier depth

For any fixed positive integer $k$, let $alpha_{k}$ denote the smallest $alpha in (0,1)$ such that the random graph sequence $left{Gleft(n, n^{-alpha}right)right}$ does not satisfy the zero-one law for the set $mathcal{E}_{k}$ of all existential first order sentences that are of quantifier depth at most $k$. This paper finds upper and lower bounds on $alpha_{k}$, showing that as $k rightarrow infty$, we have $alpha_{k} = left(k - 2 - t(k)right)^{-1}$ for some function $t(k) = Theta(k^{-2})$. We also establish the precise value of $alpha_{k}$ when $k = 4$.