Geodesic
A weak zero-one law for~sequences of random distance graphs
Sbornik. Mathematics, Tome 203 (2012) no. 7, pp. 1012-1044

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We study zero-one laws for properties of random distance graphs. Properties written in a first-order language are considered. For $p(N)$ such that $pN^{\alpha}\to\infty$ as $N\to\infty$, and $(1-\nobreak p)N^{\alpha}\to\infty$ as $N\to\infty$ for any $\alpha>0$, we succeed in refuting the law. In this connection, we consider a weak zero-one $j$-law. For this law, we obtain results for random distance graphs which are similar to the assertions concerning the classical zero-one law for random graphs. Bibliography: 18 titles.
Keywords: zero-one laws, first-order language, random graphs, distance graphs, Ehrenfeucht game. 
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     title = {A weak zero-one law for~sequences of random distance graphs},
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M. E. Zhukovskii. A weak zero-one law for~sequences of random distance graphs. Sbornik. Mathematics, Tome 203 (2012) no. 7, pp. 1012-1044. http://geodesic.mathdoc.fr/item/SM_2012_203_7_a4/