A Survey of the Holroyd-Talbot Conjecture
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909642130456576 |
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| author | Hurlbert, Glenn |
| author_facet | Hurlbert, Glenn |
| contents | A family of sets is intersecting if every pair of its members has an element in common. Such a family of sets is called a star if some element is in every set of the family. Given a graph $G$, let $μ(G)$ denote the size of the smallest maximal independent set of $G$. In 2005, Holroyd and Talbot conjectured the following generalization of the Erdős-Ko-Rado Theorem: for $1\le r\le μ(G)/2$, there is a maximum size intersecting family of independent $r$-sets that is a star. In this paper we present the history of this conjecture and survey the results that have supported it over the last 20 years. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16144 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Survey of the Holroyd-Talbot Conjecture Hurlbert, Glenn Combinatorics 05D05 (05C35, 05C05, 05C69) A family of sets is intersecting if every pair of its members has an element in common. Such a family of sets is called a star if some element is in every set of the family. Given a graph $G$, let $μ(G)$ denote the size of the smallest maximal independent set of $G$. In 2005, Holroyd and Talbot conjectured the following generalization of the Erdős-Ko-Rado Theorem: for $1\le r\le μ(G)/2$, there is a maximum size intersecting family of independent $r$-sets that is a star. In this paper we present the history of this conjecture and survey the results that have supported it over the last 20 years. |
| title | A Survey of the Holroyd-Talbot Conjecture |
| topic | Combinatorics 05D05 (05C35, 05C05, 05C69) |
| url | https://arxiv.org/abs/2501.16144 |