= Solution
Order configurations coordinatewise: $\omega\leq\omega'$ means $\omega_e\leq\omega'_e$ for every <edge>. An <increasing event> $A$ is a measurable <set> such that $\omega\in A$ and $\omega\leq\omega'$ imply $\omega'\in A$. Opening additional <edges> cannot destroy an <increasing event>. The <Harris-FKG inequality>, the <product measure> version of the <FKG inequality>, states
$$
\boxed{\mathbb P_p(A\cap B)\geq\mathbb P_p(A)\mathbb P_p(B)}
$$
for two <increasing events>. Equivalently, bounded coordinatewise increasing <random variables> $f,g$ satisfy $\mathbb E_p(fg)\geq\mathbb E_p f\,\mathbb E_p g$. The event inequality also holds for two <decreasing events>, by taking complements.
For <disjoint occurrence of increasing events>, a finite open <edge> set $K$ witnesses $A$ in $\omega$ if every configuration agreeing with $\omega$ on $K$ belongs to $A$. Since $A$ is an <increasing event> and all <edges> of $K$ are open, this says that the configuration with exactly $K$ open already forces $A$. Define $A\mathbin\square B$ to consist of configurations admitting disjoint finite witnesses $K,L$ for $A,B$. For events depending on finitely many <edges>, this is the usual disjoint-occurrence definition; it also applies to finite-connection events on the infinite <graph>. The <van den Berg-Kesten inequality> states
$$
\boxed{\mathbb P_p(A\mathbin\square B)\leq\mathbb P_p(A)\mathbb P_p(B)}.
$$
On a countable <graph>, its finite-witness version follows by taking increasing unions over finite <edge> sets. The <Harris-FKG inequality> concerns simultaneous occurrence, whereas the <van den Berg-Kesten inequality> requires separate certificates using disjoint coordinates.
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