Isomorphisms between random graphs

Consider two independent Erd\H{o}s-R\'enyi $G(N,1/2)$ graphs. We show that with probability tending to $1$ as $N\to\infty$, the largest induced isomorphic subgraph has size either $\lfloor x_N-\varepsilon_N\rfloor$ or $\lfloor x_N+\varepsilon_N \rfloor$, where $x_N=4\log_2 N -2 \log_2 \log_2 N - 2\log_2(4/e)+1$ and $\varepsilon_N = (4\log_2 N)^{-1/2}$. Using similar techniques, we also show that if $\Gamma_1$ and $\Gamma_2$ are independent $G(n,1/2)$ and $G(N,1/2)$ random graphs, then $\Gamma_2$ contains an isomorphic copy of $\Gamma_1$ as an induced subgraph with high probability if $n\le \lfloor y_N - \varepsilon_N \rfloor$ and does not contain an isomorphic copy of $\Gamma_1$ as an induced subgraph with high probability if $n>\lfloor y_N+\varepsilon_N \rfloor$, where $y_N=2\log_2 N+1$ and $\varepsilon_N$ is as above.