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arXiv · 2507.23375

Recent advances in arrow relations and traces of sets

Mingze Li·Jie Ma·Mingyuan Rong
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Abstract

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.

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Mingze Li, Jie Ma, Mingyuan Rong. 2025-07-31. Recent advances in arrow relations and traces of sets. https://arxiv.org/abs/2507.23375

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