When do the Kahn-Kalai bounds provide nontrivial information?

• Published in: Journal of inequalities and applications Vol. 2025; no. 1; pp. 23 - 7
• Main Authors: Christopherson, Bryce Alan, Baretz, Jack
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 26.02.2025; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Kahn-Kalai; Mathematics; Mathematics and Statistics; Park-Pham; Theorems; Thresholds; Upper bounds; 60C05; 68R01; Thresholds; Park-Pham; 05C80; Kahn-Kalai; 06A07
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-025-03272-z

The Park-Pham theorem (previously known as the Kahn-Kalai conjecture) bounds the critical probability, p c ( F ) , of the a nontrivial property F ⊆ 2 X that is closed under supersets by the product of a universal constant K , the expectation threshold of the property, q ( F ) , and the logarithm of the size of the property’s largest minimal element, log ℓ ( F ) . That is, the Park-Pham theorem asserts that p c ( F ) ≤ K q ( F ) log ℓ ( F ) . Since the critical probability p c ( F ) always satisfies p c ( F ) < 1 , one may ask when the upper bound posed by Kahn and Kalai gives us more information than this–that is, when is it true that K q ( F ) log ℓ ( F ) < 1 ? In this short note, we provide a number of necessary conditions for this to happen and give a few sufficient conditions for the bounds to provide new (and, in fact, asymptotically perfect) information along the way. In the most interesting case where ℓ ( F n ) → ∞ , we prove the following relatively strong necessary condition for the Kahn-Kalai bounds to provide nontrivial information: For every positive integer t , every collection of all-but- t of the minimal elements of F n may have nonempty intersection for only finitely many n . Consequently, not only must the number of minimal elements become arbitrarily large, but so too must the size of any cover. Intuitively, this means that such sequences F n must occupy an ever-widening ‘wedge’ in 2 X n : the further F n climbs up 2 X n in one area, the further it must spread down and across 2 X n in another.