Large components in random induced subgraphs of n-cubes

In this paper we study random induced subgraphs of the binary $n$-cube, $Q_2^n$. This random graph is obtained by selecting each $Q_2^n$-vertex with independent probability $lambda_n$. Using a novel construction of subcomponents we study the largest component for $lambda_n=frac{1+chi_n}{n}$, where $epsilonge chi_nge n^{-{1/3}+ delta}$, $delta>0$. We prove that there exists a.s. a unique largest component $C_n^{(1)}$. We furthermore show that $chi_n=epsilon$, $| C_n^{(1)}|sim alpha(epsilon) frac{1+chi_n}{n} 2^n$ and for $o(1)=chi_nge n^{-{1/3}+delta}$, $| C_n^{(1)}| sim 2 chi_n frac{1+chi_n}{n} 2^n$ holds. This improves the result of cite{Bollobas:91} where constant $chi_n=chi$ is considered. In particular, in case of $lambda_n=frac{1+epsilon} {n}$, our analysis implies that a.s. a unique giant component exists.