  We consider two variants of the induced subgraph isomorphism problem for two
independent binomial random graphs with constant edge-probabilities p_1,p_2. We
resolve several open problems of Chatterjee and Diaconis, and also confirm
simulation-based predictions of McCreesh, Prosser, Solnon and Trimble: (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 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.
