Thresholds vs. expectation thresholds for non-spanning graphs
Quentin Dubroff
TL;DR
This work investigates thresholds for containing a fixed graph $H$ in the Erdős–Rényi graph $G_{n,p}$ and its fractional counterpart $q_f(H)$. It demonstrates that even small graphs (not necessarily nearly spanning) can exhibit a substantial gap between $p_c(H)$ and $q_f(H)$, by constructing, for every $m\le n$, a graph with $v_H=m$ and $p_c(H) > c\,q_f(H)\log^{1/2}(v_H)$. The construction uses a gadget built from a random $4$-regular graph on $t$ vertices plus a pairwise attachment scheme, yielding $v_H=t+\binom{t}{2}$ and $e_H=2v_H$, and it shows that the fractional threshold $q_f(H)$ is $O(1/\sqrt{n})$ whp while the actual threshold is inflated by a $\log^{1/2}$ factor. The main technical contribution is the precise analysis of the threshold via the fractional method and a coupon-collector-like prerequisite, supported by BK-type negative correlation arguments and careful counting of extensions, establishing both lower and upper bounds that are tight up to constants.
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 $p_c(H) > q_f(H) c \log^{1/2}(v_H).$