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 K n , p ( n ). Fix a strictly strongly balanced graph G . A subgraph of K n, p ( n) isomorphic to G is called a G ‐subgraph. Let X n be the number of G ‐subgraphs of K n, p ( n) vertex disjoint to all other G ‐subgraphs. We show that if E[ X n ]→∞ as n →, then X n /E[ X n ] converges to 1 in probability. Also, if E[ X n ]→ c as n →∞, then X n 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].