Recent advances in arrow relations and traces of sets

TL;DR

This survey reviews recent progress on arrow relations in extremal set theory, focusing on how large families of sets guarantee the existence of certain traces with many distinct subsets.

Contribution

It provides a comprehensive overview of recent advances and diverse extremal perspectives on arrow relations and traces in set theory.

Findings

Summarizes recent results on arrow relations.

Highlights extremal set theory perspectives.

Connects various problems and solutions in the field.

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.