Year: 2026
Type: preprint
Abstract: We show that the uniform probability measure on copies of a nonempty graph $H$ in $K_n$ is $Cq_H\log(2e(H))$-spread, where $q_H$ is its graphic expectation threshold. Consequently, the fractional expectation threshold of $H$ is at most $Cq_H\log(2e(H))$. We remove the logarithmic factor for trees and for graphs whose average degree is at least the logarithm of their maximum degree. This proves the ``second'' Kahn-Kalai conjecture for these two classes, which encompass most of the standard families studied in random graph containment problems.
Source: arXiv (Cornell University)
Author: Tran, Tuan
Institution: University of Science and Technology of China
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213580361