Thresholds vs. expectation thresholds for non-spanning graphs
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918316457590784 |
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| author | Dubroff, Quentin |
| author_facet | Dubroff, Quentin |
| contents | 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 $p_c(H) > q_f(H) c \log^{1/2}(v_H).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_00278 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Thresholds vs. expectation thresholds for non-spanning graphs Dubroff, Quentin Combinatorics 05C80 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 $p_c(H) > q_f(H) c \log^{1/2}(v_H).$ |
| title | Thresholds vs. expectation thresholds for non-spanning graphs |
| topic | Combinatorics 05C80 |
| url | https://arxiv.org/abs/2602.00278 |