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A dimension-free comparison between expectation thresholds and fractional expectation thresholds

Year: 2026

Type: preprint

Abstract: We prove a dimension-free comparison between the expectation threshold $q(\mathcal F)$ and the fractional expectation threshold $q_f(\mathcal F)$ for any nontrivial increasing family $\mathcal F$ on a finite ground set. Specifically, we show that there is a universal constant $K>0$ such that \[ q_f(\mathcal F)\le Kq(\mathcal F)\max\{1,\log\log(1/q(\mathcal F))\}. \] Combining this comparison with a recent result of Li [arXiv:2609.08967] on the fractional version of Talagrand's discrete Convexity Conjecture, we obtain a dimension-independent bound toward the conjecture.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.14681

Author: Park, Jinyoung

Institutions: Courant Institute of Mathematical Sciences, New York University

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funder: National Science Foundation

Awards: CAREER: Threshold Phenomena for Random Discrete Structures and Asymptotic Enumeration Problems, G6671297155

OpenAlex ID: W7213280207

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