Start unconditionally
Use the Park–Pham lower bound
The original theorem supplies information about the probability of an increasing property under product measure.
Paper project · Combinatorics and probability
Conditional Park–Pham Bounds under Positive Correlation
Overview
Conditioning need not destroy a threshold bound.
The Park–Pham theorem controls the probability of a monotone property under product measure. After conditioning on another event, that control can disappear unless the conditioning interacts favorably with the target property.
The paper isolates a clean sufficient condition: if the target property and conditioning event are positively correlated, then the conditional probability is at least the original probability, so the usual lower bound transfers. Increasing conditioning events follow from Harris’s inequality; independence gives equality even for certain nonmonotone events.
Check correlation
Compare F∩B with F and B
Positive correlation gives P(F|B)≥P(F), which is exactly the inequality needed to retain the unconditional lower bound.
Move beyond the cube
Embed ordered settings
The same transfer idea is formulated for finite posets embedded in Boolean lattices and illustrated with permutation-pattern upper sets.
Exact conditioning laboratory
Correlation controls what survives conditioning.
Explore all 16 subsets of a four-element set under product measure. Three exact conditioning events show positive correlation, nonmonotone independence, and failure under an unfavorable condition.
Citation
Use or reference this work.
The manuscript was first posted in 2024 and revised in 2026; the citation below uses the arXiv record.
BibTeX citation
@article{christopherson2024conditional,
title = {Conditional Park--Pham Bounds under Positive Correlation},
author = {Christopherson, Bryce Alan and Colgrove, Darian},
year = {2024},
eprint = {2402.17872},
archivePrefix = {arXiv},
primaryClass = {math.CO},
url = {https://arxiv.org/abs/2402.17872}
}