We consider two variants of the induced subgraph isomorphism problem for two independent binomial random graphs with constant edge-probabilities \(p_1,p_2\) . In particular, (i) we prove a sharp threshold result for the appearance of \(G_{n,p_1}\) as an induced subgraph of \(G_{N,p_2}\) , (ii) we show two-point concentration of the size of the maximum common induced subgraph of \(G_{N, p_1}\) and \(G_{N,p_2}\) , and (iii) we show that the number of induced copies of \(G_{n,p_1}\) in \(G_{N,p_2}\) has an unusual limiting distribution. These results confirm simulation-based predictions of McCreesh, Prosser, Solnon and Trimble, and resolve several open problems of Chatterjee and Diaconis. The proofs are based on careful refinements of the first and second moment method, using extra twists to (a) take some non-standard behaviors into account, and (b) work around the large variance issues that prevent standard applications of these methods.
