On the probability of the occurrence of a copy of a fixed graph in a random distance graph

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. Erd?s 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.