A correlation inequality and a poisson limit theorem for nonoverlapping balanced subgraphs of a random graph

We consider non‐overlapping subgraphs of fixed order in the random graph Kn, p(n). Fix a strictly strongly balanced graph G. A subgraph of Kn, p(n) isomorphic to G is called a G‐subgraph. Let Xn be the number of G‐subgraphs of Kn, p(n) vertex disjoint to all other G‐subgraphs. We show that if E[Xn]→∞ as n→, then Xn/E[Xn] converges to 1 in probability. Also, if E[Xn]→c as n→∞, then Xn satisfies a Poisson limit theorem. the Poisson limit theorem is shown using a correlation inequality similar to those appeared in Janson, Łuczak, and Ruciñski[8] and Boppana and Spencer [4].