Note on a conjecture of Talagrand: expectation thresholds vs. fractional expectation thresholds

TL;DR

This paper explores a restricted version of Talagrand's conjecture relating expectation thresholds and fractional expectation thresholds, deriving it from a proven conjecture on selector processes and utilizing recent quantitative improvements.

Contribution

It demonstrates that a restricted form of Talagrand's conjecture follows from a strengthened selector process conjecture, connecting two significant conjectures in probabilistic combinatorics.

Findings

Restricted conjecture follows from selector process conjecture

Utilizes recent quantitative strengthening by Bednorz, Martynek, and Meller

Connects expectation thresholds with fractional expectation thresholds

Abstract

We show that a restricted version of a conjecture of M. Talagrand on the relation between "expectation thresholds" and "fractional expectation thresholds" follows easily from a strong version of a second conjecture of Talagrand, on "selector processes." The selector process conjecture was proved by Park and Pham, and the quantitative strengthening used here is due to Bednorz, Martynek, and Meller.