在正相关条件下的Park-Pham界限
Conditional Park--Pham Bounds under Positive Correlation
AI总结：
本文研究了在正相关条件下单调性质与条件事件之间的Park-Pham下界转换问题，并探讨了有限偏序集嵌入布尔格中的转移原理。
摘要：
我们记录了ε依赖的Park-Pham定理的条件形式。如果一个单调性质F⊆2^X在乘积测度μ_p下与条件事件B⊆2^X正相关，那么通常的Park-Pham下界对于μ_p(F)会转移到条件概率P(X_p∈F|X_p∈B)。这特别给出了通过Harris不等式对递增条件事件的条件Park-Pham界限，以及对与目标性质无关的非单调条件事件的界限。我们还为嵌入在布尔格中的有限偏序集制定了转移原理，并通过排列类中的模式包含上界进行了说明。
英文摘要：
We record a conditional form of the $ε$-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 $μ_p$, then the usual Park--Pham lower bound for $μ_p(\mathcal{F})$ 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.
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