Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 209 / 1 / d2 / Solution

If an infinite cluster $C$ avoids $B$, the set of first coordinates of its vertices has a least value $n_0\ge 1$. Then $C$, viewed in $H_{n_0}$, intersects $B_{n_0}$. Repeated application of part (d1) embeds it in an infinite cluster of $H_{n_0-1}$ meeting $B_{n_0-1}$, eventually producing an infinite cluster of $H$ meeting $B$. Because each enlarged cluster contains $C$, already the first step contradicts the minimality of $n_0$. Thus such a cluster has probability zero.