Large components in random induced subgraphs of -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 λ n . Using a novel construction of subcomponents we study the largest component for λ n = 1 + χ n n , where ϵ ≥ χ n ≥ n − 1 3 + δ , δ > 0 . We prove that there exists a.s. a unique largest component C n ( 1 ) . We furthermore show that for χ n = ϵ , we have | C n ( 1 ) | ∼ α ( ϵ ) 1 + χ n n 2 n and for o ( 1 ) = χ n ≥ n − 1 3 + δ , | C n ( 1 ) | ∼ 2 χ n 1 + χ n n 2 n holds. This improves the result of [B. Bollobás, Y. Kohayakawa, T. Luczak, On the evolution of random boolean functions, Extremal Problems Finite Sets (1991) 137–156] where constant χ n = χ is considered. In particular, in the case of λ n = 1 + ϵ n , our analysis implies that a.s. a unique giant component exists.