Structure and properties of large intersecting families

TL;DR

This paper investigates the structure and extremal properties of large intersecting families of sets, providing new theorems and generalizations related to Erdős–Ko–Rado, Hilton–Milner, and Erdős Matching conjectures.

Contribution

It offers a conclusive version of Frankl's theorem, extends degree and subset degree versions of key theorems, and generalizes structural results for large intersecting families.

Findings

Proved a conclusive version of Frankl's theorem on bounded maximal degree.

Extended the degree version of the Erdős–Ko–Rado and Hilton–Milner theorems.

Generalized structural theorems for large intersecting families.

Abstract

We say that a family of $k$-subsets of an $n$-element set is intersecting, if any two of its sets intersect. In this paper we study different extremal properties of intersecting families, as well as the structure of large intersecting families. We also give some results on $k$-uniform families without $s$ pairwise disjoint sets, related to the Erd\H{o}s Matching Conjecture. We prove a conclusive version of Frankl's theorem on intersecting families with bounded maximal degree. This theorem, along with its generalizations to cross-intersecting families, implies many results on the topic, obtained by Frankl, Frankl and Tokushige, Kupavskii and Zakharov and others. We study the structure of large intersecting families, obtaining some general structural theorems which generalize the results of Han and Kohayakawa, as well as Kostochka and Mubayi. We give degree and subset degree version of the Erd\H{o}s--Ko--Rado and the Hilton--Milner theorems, extending the results of Huang and Zhao, and Frankl, Han, Huang and Zhao. We also extend the range in which the degree version of the Erd\H{o}s Matching conjecture holds.