A Conditional Extension of the Park-Pham Theorem

Neglecting many motivating details for the Park-Pham theorem (previously known as the Kahn-Kalai conjecture), the result starts with a finite set $X$, a non-trivial upper set $\mathcal{F} \subseteq 2^X$, and a particular parameterized family of random variables $X_p$, then proceeds to provide an upper bound on the value $p_c(\mathcal{F})$ such that $\mathbb{P}(X_{p_c(\mathcal{F})} \in \mathcal{F}) = 1/2$. A natural question to ask is if there is an analog to the Park-Pham theorem for upper sets in finite posets other than $2^X$ and other parameterized families of random variables taking values in them. In this short note, we show that there is, with minor adjustments, in at least one circumstance. This is done by producing a conditional version of the $\epsilon$-dependent form of the Park-Pham theorem, which has practical implications for the study of large neural networks and may also be interesting in its own right.