Conditional Park--Pham Bounds under Positive Correlation

We record a conditional form of the $\epsilon$-dependent Park–Pham theorem. If a monotone property $\mathcal{F}\subseteq 2^X$ is positively correlated with a conditioning event $B\subseteq 2^X$ under the product measure ${\mu }_p$, then the usual Park–Pham lower bound for ${\mu }_p(\mathcal{F})$ transfers to the conditional probability $\mathbb{P}(X_p\in \mathcal{F}\mid X_p\in B)$. This gives, in particular, conditional Park–Pham bounds for increasing conditioning events by Harris’s inequality, and for nonmonotone conditioning events that are independent of the target property. We also formulate the transfer principle for finite posets embedded in Boolean lattices and illustrate it with pattern-containment upper sets in permutation classes.