The Median of the Number of Simple Paths on Three Vertices in the Random Graph

We study the asymptotic behavior of the random variable equal to the number of simple paths on three vertices in the binomial random graph in which the edge probability equals the threshold probability of the appearance of such paths. We prove that, for any fixed nonnegative integer $b$ and a sufficiently large number $n$ of vertices of the graph, the probability that the number of simple paths on three vertices in the given random graph is $b$ decreases with $n$. As a consequence of this result, we obtain the median of the number of simple paths on three vertices for sufficiently large $n$.