Structure of non-trivial intersecting families

TL;DR

This paper investigates the structure of large intersecting families of k-subsets of an n-set, extending previous results to provide a comprehensive understanding of their configuration.

Contribution

It generalizes and completes earlier structural results on large intersecting families, offering a conclusive characterization.

Findings

Extended previous structural results

Provided a comprehensive classification of large intersecting families

Unified earlier partial results into a complete framework

Abstract

We say that a family of $k$-subsets of an $n$-element set is {\it intersecting}, if any two of its sets intersect. In this paper, we study the structure of large intersecting families. Several years ago, Han and Kohayakawa (Proc. AMS, 2017), and then Kostochka and Mubayi (Proc. AMS, 2017) obtained certain structural results concerning large intersecting families. In this paper, we extend and generalize their results, giving them a conclusive form.