Intersecting families of sets are typically trivial
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2021
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| _version_ | 1866917812756283392 |
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| author | Balogh, József Garcia, Ramon I. Li, Lina Wagner, Adam Zsolt |
| author_facet | Balogh, József Garcia, Ramon I. Li, Lina Wagner, Adam Zsolt |
| contents | A family of subsets of $[n]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran independently showed that for $n\geq 2k + c\sqrt{k\ln k}$, almost all $k$-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for $n\geq 2k+ 100\ln k$. Our proof uses, among others, Sapozhenko's graph container lemma and the Das-Tran removal lemma. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_03260 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Intersecting families of sets are typically trivial Balogh, József Garcia, Ramon I. Li, Lina Wagner, Adam Zsolt Combinatorics A family of subsets of $[n]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran independently showed that for $n\geq 2k + c\sqrt{k\ln k}$, almost all $k$-uniform intersecting families are stars. Improving their result, we show that the same conclusion holds for $n\geq 2k+ 100\ln k$. Our proof uses, among others, Sapozhenko's graph container lemma and the Das-Tran removal lemma. |
| title | Intersecting families of sets are typically trivial |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2104.03260 |