The largest critical point in the zero-one $k$-law

The largest value of $\alpha<1$ is established for which the random graph $G(n,n^{-\alpha})$ does not obey the zero-one $k$-law for first-order properties. As was already known, the zero-one $k$-law is valid for all $\alpha>1-1/(2^{k}-2)$ with the exception of $1-1/(2^{k}-1)$, $1-1/2^{k}$. For $\alpha=1-1/(2^k-2)$ the law fails. In this work, it is shown that the law is valid for $\alpha\in\{1-1/(2^{k}-1),1-1/2^{k}\}$.
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