Test nontriviality
Compare with the universal bound pc<1
The question is not whether the theorem is true, but whether its numerical upper bound contributes any new information.
Paper project · Combinatorics and probability
When do the Kahn–Kalai bounds provide nontrivial information?
Overview
A powerful bound can still be vacuous.
The Park–Pham theorem bounds the critical probability pc(F) of a nontrivial upper set using its expectation threshold q(F) and the size ℓ(F) of its largest minimal element. Because pc(F)<1 is always known, the theorem is informative only when Kq(F)logℓ(F)<1.
The paper studies when that happens. In the regime ℓ(Fn)→∞, it proves a strong necessary dispersal condition: after deleting any fixed number of minimal elements, the remaining collection can have a common intersection only finitely often. It also identifies sufficient conditions under which the bound becomes asymptotically exact.
Inspect minimal elements
Shared cores obstruct useful bounds
If almost all minimal elements keep a persistent common intersection, the Park–Pham estimate cannot become nontrivial in the growing regime.
Look for a widening wedge
Growth must be accompanied by spread
The minimal elements must proliferate and disperse through the Boolean lattice rather than climbing inside a narrow principal region.
Boolean-lattice geometry
Useful bounds require more than height.
The paper’s geometric intuition is that a growing upper set must spread down and across the lattice as it climbs; a persistent shared core makes the bound vacuous.
minimal elements
their upper closure
A single lattice shows height and spread at one scale. The sequence below shows why both must grow together.
In a Hasse diagram, the region occupied by an upper set must be “wedge-shaped” and never too skinny: the farther it climbs in one area, the farther it must spread down and across in another.
stage 1 · max rank 48 minimal generators8 new · ℓ(F)=4
stage 2 · max rank 620 minimal generators8 retained · 12 new · ℓ(F)=6
stage 3 · max rank 938 minimal generators20 retained · 18 new · ℓ(F)=9
Citation
Use or reference this work.
The article is published open access; the arXiv version is also linked above.
BibTeX citation
@article{christopherson2025kahnkalai,
title = {When do the Kahn--Kalai bounds provide nontrivial information?},
author = {Christopherson, Bryce Alan and Baretz, Jack},
journal = {Journal of Inequalities and Applications},
volume = {2025},
number = {1},
eid = {23},
pages = {1--7},
year = {2025},
doi = {10.1186/s13660-025-03272-z}
}