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Some results on fractional vs. expectation thresholds

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arxiv 2311.08163 v1 pith:UO3Q44SU submitted 2023-11-14 math.CO

classification math.CO
keywords expectationfractionalrespsomeconjecturefactormathcalpark
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abstract

A conjecture of Talagrand (2010) states that the so-called expectation and fractional expectation thresholds are always within at most some constant factor from each other. Expectation (resp. fractional expectation) threshold $q$ (resp. $q_f$) for an increasing nontrivial class $\mathcal{F}\subseteq 2^X$ allows to locate the threshold for $\mathcal{F}$ within a logarithmic factor (these are important breakthrough results of Park and Pham (2022), resp. Frankston, Kahn, Narayanan and Park (2019)). We will survey what is known about the relation between $q$ and $q_f$ and prove some further special cases of Talagrand's conjecture.

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  1. Further remarks on fractional vs. expectation thresholds

    math.CO 2025-05 conditional novelty 6.0 of 10

    The authors verify Talagrand's threshold conjecture for the family of ℓ-uniform hypergraph cliques, with a uniform constant L = 2^12 e^16 for large ground sets.

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