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The probability of nonexistence of a subgraph in a moderately sparse random graph

We develop a general procedure that finds recursions for statistics counting isomorphic copies of a graph $G_0$ in the common random graph models ${\cal G}(n,m)$ and ${\cal G}(n,p)$. Our results apply when the average degrees of the random graphs are below the threshold at which each edge is included in a copy of $G_0$. This extends an argument given earlier by the second author for $G_0=K_3$ with a more restricted range of average degree. For all strictly balanced subgraphs $G_0$, our results gives much information on the distribution of the number of copies of $G_0$ that are not in large "clusters" of copies. The probability that a random graph in ${\cal G}(n,p)$ has no copies of $G_0$ is shown to be given asymptotically by the exponential of a power series in $n$ and $p$, over a fairly wide range of $p$. A corresponding result is also given for ${\cal G}(n,m)$, which gives an asymptotic formula for the number of graphs with $n$ vertices, $m$ edges and no copies of $G_0$, for the applicable range of $m$. An example is given, computing the asymptotic probability that a random graph has no triangles for $p=o(n^{-7/11})$ in ${\cal G}(n,p)$ and for $m=o(n^{15/11})$ in ${\cal G}(n,m)$, extending results of the second author.