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A dimension-free comparison between expectation thresholds and fractional expectation thresholds
Abstract
We prove a dimension-free comparison between the expectation threshold q(ℱ) and the fractional expectation threshold qf(ℱ) for any nontrivial increasing family ℱ on a finite ground set. Specifically, we show that there is a universal constant K>0 such that qf(ℱ)≤ Kq(ℱ)max{1,loglog(1/q(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.
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BibTeX
@misc{park2026dimensionfree,
title = {{A dimension-free comparison between expectation thresholds and fractional expectation thresholds}},
author = {Jinyoung Park},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14681},
eprint = {2609.14681},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14681},
}
RIS
TY - UNPB
TI - A dimension-free comparison between expectation thresholds and fractional expectation thresholds
AU - Jinyoung Park
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14681
UR - https://arxiv.org/abs/2609.14681
AB - We prove a dimension-free comparison between the expectation threshold q(ℱ) and the fractional expectation threshold q_f(ℱ) for any nontrivial increasing family ℱ on a finite ground set. Specifically, we show that there is a universal constant K>0 such that q_f(ℱ)≤ Kq(ℱ)max{1,loglog(1/q(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.
ER -