Intersecting families of sets and permutations: a survey

TL;DR

This survey reviews key results, conjectures, and open problems in extremal set theory related to $t$-intersecting families of sets and permutations across various important families, highlighting recent advances and extensions.

Contribution

It compiles and discusses known results, conjectures, and open problems in the study of $t$-intersecting families for multiple combinatorial structures, providing a comprehensive overview.

Findings

Summarizes major theorems and conjectures in $t$-intersecting families.

Highlights extensions and consequences of existing results.

Identifies open problems and directions for future research.

Abstract

A family $\mathcal{A}$ of sets is said to be \emph{$t$-intersecting} if any two sets in $\mathcal{A}$ have at least $t$ common elements. A central problem in extremal set theory is to determine the size or structure of a largest $t$-intersecting sub-family of a given family $\mathcal{F}$. We give a survey of known results, conjectures and open problems for various important families $\mathcal{F}$, namely, power sets, levels of power sets, hereditary families, families of signed sets, families of labeled sets, and families of permutations. We also provide some extensions and consequences of known results.