On Connection Probability of a Random Subgraph of an $n$-Dimensional Cube

The following random procedure is considered. Each edge of an $n$-dimensional cube is removed with a specified probability, independently of the remaining edges. In the paper it is shown that if the probability of removing an edge $q$ has been fixed, while the dimensionality of the cube tends to infinity, then the probability that the graph formed by the unremoved edges is connected tends to 1 when $q<1/2$ and to 0 when $q>1/2$. By virtue of the upper bound obtained in [M. V. Lomonosov and V. P. Polesskii, Probl. Peredachi Inf., 7, No. 4, 78–81 (1971)] for the connection probability of a random graph, this assertion proves the asymptotic optimality in the sense of reliability of the graph of an $n$-dimensional cube among the graphs with the same number of vertices and edges.