Thresholds and expectation thresholds for larger p
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917206380511232 |
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| author | Przybyłowski, Tomasz Riordan, Oliver |
| author_facet | Przybyłowski, Tomasz Riordan, Oliver |
| contents | 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 $p_\mathrm{c} \leqslant K q_\mathrm{c} \log_2 l$ for some universal constant $K$. Here we slightly strengthen their result by showing that $p_\mathrm{c} \leqslant 1 - \mathrm{e}^{-K q_\mathrm{c} \log_2 l}$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_03327 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Thresholds and expectation thresholds for larger p Przybyłowski, Tomasz Riordan, Oliver Combinatorics Probability 60C05 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 $p_\mathrm{c} \leqslant K q_\mathrm{c} \log_2 l$ for some universal constant $K$. Here we slightly strengthen their result by showing that $p_\mathrm{c} \leqslant 1 - \mathrm{e}^{-K q_\mathrm{c} \log_2 l}$. 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. |
| title | Thresholds and expectation thresholds for larger p |
| topic | Combinatorics Probability 60C05 |
| url | https://arxiv.org/abs/2302.03327 |