Geodesic
On the Probability of the Occurrence of a Copy of a Fixed Graph in a Random Distance Graph
Matematičeskie zametki, Tome 92 (2012) no. 6, pp. 844-855

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The threshold probability of the occurrence of a copy of a balanced graph in a random distance graph is obtained. The technique used by P. Erdős and A. Rényi for determining the threshold probability for the classical random graph could not be applied in the model under consideration. In this connection, a new method for deriving estimates of the number of copies of a balanced graph in a complete distance graph is developed.
Keywords: random distance graph, complete distance graph, balanced graph, threshold probability, Erdős–Rényi theorem on threshold probability, Stirling's formula. 
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     author = {M. E. Zhukovskii},
     title = {On the {Probability} of the {Occurrence} of a {Copy} of a {Fixed} {Graph} in a {Random} {Distance} {Graph}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {844--855},
     publisher = {mathdoc},
     volume = {92},
     number = {6},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2012_92_6_a4/}
}
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M. E. Zhukovskii. On the Probability of the Occurrence of a Copy of a Fixed Graph in a Random Distance Graph. Matematičeskie zametki, Tome 92 (2012) no. 6, pp. 844-855. http://geodesic.mathdoc.fr/item/MZM_2012_92_6_a4/