  For any fixed positive integer $k$, let $\alpha_{k}$ denote the smallest
$\alpha \in (0,1)$ such that the random graph sequence $\left\{G\left(n,
n^{-\alpha}\right)\right\}$ 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 $k = 4$.
