Let $p_\mathrm{c}$ and $q_\mathrm{c}$ be the threshold and the expectation threshold, respectively, of an increasing family $\mathcal{F}$ of subsets of a finite set $X$, and let $l$ be the size of a largest minimal element of $\mathcal{F}$. Recently, Park and Pham proved the Kahn–Kalai conjecture, which says that $p_\mathrm{c} \le K q_\mathrm{c} \log_2 l$ for some universal constant $K$. Here, we slightly strengthen their result by showing that $p_\mathrm{c} \le 1 - \mathrm{e}^{-K q_\mathrm{c} \log_2 l}$. The idea is to apply the Park-Pham Theorem to an appropriate "cloned" family $\mathcal{F}_k$, reducing the general case (of this and related results) to the case where the individual element probability $p$ is small.
@article{10_37236_12825,
author = {Tomasz Przyby{\l}owski and Oliver Riordan},
title = {Thresholds, expectation thresholds and cloning},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {4},
doi = {10.37236/12825},
zbl = {1558.60026},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12825/}
}
TY - JOUR
AU - Tomasz Przybyłowski
AU - Oliver Riordan
TI - Thresholds, expectation thresholds and cloning
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/12825/
DO - 10.37236/12825
ID - 10_37236_12825
ER -
%0 Journal Article
%A Tomasz Przybyłowski
%A Oliver Riordan
%T Thresholds, expectation thresholds and cloning
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/12825/
%R 10.37236/12825
%F 10_37236_12825
Tomasz Przybyłowski; Oliver Riordan. Thresholds, expectation thresholds and cloning. The electronic journal of combinatorics, Tome 31 (2024) no. 4. doi: 10.37236/12825
