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Some results on fractional vs. expectation thresholds
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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.
Forward citations
Cited by 3 Pith papers
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A sharp version of Talagrand's selector process conjecture and an application to rounding fractional covers
Proves a sharp selector-process theorem and uses it to show that any fractional cover supported on sets of size at most t can be rounded to an integral cover at density cp/log t, resolving Talagrand's conjecture for b...
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Further remarks on fractional vs. expectation thresholds
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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Note on a conjecture of Talagrand: expectation thresholds vs. fractional expectation thresholds
For any fixed set-size bound r, a fractional threshold implies a genuine expectation threshold with a constant that depends on r, a restricted form of Talagrand's conjecture.
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