Around the Complete Intersection Theorem

TL;DR

This survey reviews the Complete Intersection Theorem by Ahlswede and Khachatrian, discussing its historical development and various applications in combinatorics.

Contribution

It provides a comprehensive overview of the theorem's history, significance, and practical applications in the field.

Findings

Highlights the theorem's role in extremal set theory

Summarizes key applications in combinatorics

Connects the theorem to related mathematical results

Abstract

The celebrated theorem of Ahlswede and Khachatrian determines the maximum size of a family of $k$-element subsets of an $n$-element set where the intersection of any two subsets has at least $r$ elements. This survey first gives a historical introduction, then it shows some applications.