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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.23375 |
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| _version_ | 1866911085414580224 |
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| author | Li, Mingze Ma, Jie Rong, Mingyuan |
| author_facet | Li, Mingze Ma, Jie Rong, Mingyuan |
| contents | The arrow relation, a central concept in extremal set theory, captures quantitative relationships between families of sets and their traces. Formally, the arrow relation $(n, m) \rightarrow (a, b)$ signifies that for any family $\mathcal{F} \subseteq 2^{[n]}$ with $|\mathcal{F}| \geqslant m$, there exists an $a$-element subset $T \subseteq [n]$ such that the trace $\mathcal{F}_{|T} = \{ F \cap T : F \in \mathcal{F} \}$ contains at least $b$ distinct sets. This survey highlights recent progress on a variety of problems and results connected to arrow relations. We explore diverse topics, broadly categorized by different extremal perspectives on these relations, offering a cohesive overview of the field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23375 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recent advances in arrow relations and traces of sets Li, Mingze Ma, Jie Rong, Mingyuan Combinatorics The arrow relation, a central concept in extremal set theory, captures quantitative relationships between families of sets and their traces. Formally, the arrow relation $(n, m) \rightarrow (a, b)$ signifies that for any family $\mathcal{F} \subseteq 2^{[n]}$ with $|\mathcal{F}| \geqslant m$, there exists an $a$-element subset $T \subseteq [n]$ such that the trace $\mathcal{F}_{|T} = \{ F \cap T : F \in \mathcal{F} \}$ contains at least $b$ distinct sets. This survey highlights recent progress on a variety of problems and results connected to arrow relations. We explore diverse topics, broadly categorized by different extremal perspectives on these relations, offering a cohesive overview of the field. |
| title | Recent advances in arrow relations and traces of sets |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.23375 |