Structure and properties of large intersecting families

TL;DR

This paper investigates the structure and size of large intersecting families of sets, extending classical theorems and providing new bounds and structural insights for various intersecting set family problems.

Contribution

It proves a comprehensive version of Frankl's theorem with bounded maximal degree and extends classical intersecting family results to broader conditions and parameters.

Findings

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

Extends classical results to large intersecting families with covering number 3.

Provides structural theorems for large intersecting families and bounds on their sizes.

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 properties and structure of large intersecting families. 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, strengthens the results obtained by Frankl, Frankl and Tokushige, Kupavskii and Zakharov and others. We study the structure of large intersecting families, obtaining some very general structural theorems which extend the results of Han and Kohayakawa, as well as Kostochka and Mubayi. We also obtain an extension of some classic problems on intersecting families introduced in the 70s. We extend an old result of Frankl, in which he determined the size and structure of the largest intersecting family of $k$-sets with covering number $3$ for $n>n_0(k)$. We obtain the same result for $n>Ck$, where $C$ is an absolute constant. Finally, we obtain a similar extension for the following problem of Erd\H os, Rothschild and Sz\'{e}meredi: what is the largest intersecting family, in which no element is contained in more than a $c$-proportion of the sets, for different values of $c$.