Thresholds vs. expectation thresholds for non-spanning graphs

TL;DR

This paper explores the relationship between the threshold probability for a graph to appear in a random graph and the fractional expectation threshold, constructing small graphs where these thresholds differ significantly.

Contribution

It introduces new small graphs demonstrating that the threshold for containment can differ from the fractional expectation threshold by a logarithmic factor.

Findings

Constructed small graphs with threshold significantly larger than their fractional expectation threshold.

Showed that for any size, there exist graphs where the threshold exceeds the fractional expectation threshold by a factor involving ^{1/2}log^{1/2}(v_H).

Challenged previous assumptions that large gaps only occur for graphs with more than half the vertices.

Abstract

The threshold $p_c(H)$ for the event that the binomial random graph $G_{n,p}$ contains a copy of a graph $H$ is the unique $p$ for which $\mathbb{P}(H \subseteq G_{n,p}) = 1/2$, and the fractional expectation threshold $q_f(H)$ is roughly the best lower bound on $p_c(H)$ using simple expectation considerations. All previously known $H$'s with $p_c(H)$ substantially larger than $q_f(H)$ have the property that $v_H > n/2$ (where $v_H$ is the number of vertices of $H$). We construct small graphs whose threshold for containment in $G_{n,p}$ is of different order than their corresponding fractional expectation threshold: there is a constant $c > 0$ such that for any $m \; (\leq n)$, there is a graph $H$ with $v_H = m$ and $p_c(H) > q_f(H) c \log^{1/2}(v_H).$