Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 204 / 3 / e / Solution

Take $A=\{N\leftrightarrow S\}$ and let $s=1-p\downarrow 0$ withThen $sq\to \infty$, $s^{2}q\to 0$, and $s^4{q}^3=s^{-1/2}\to \infty$. Part d gives ${\mathbb{P}}_{\mathrm{FK}}^{p,q}(A)\to 0$.

On a four-cycle, $A\square A$ occurs exactly when all four edges are open, because the two length-two paths from $N$ to $S$ are the only disjoint witnesses. HenceChoosing $s$ sufficiently small gives the two numerical inequalities in the question.