Regular bipartite graphs and intersecting families

TL;DR

This paper introduces a unified approach using regular bipartite graphs to prove key theorems in intersecting families, simplifying proofs and deriving new stronger results in extremal set theory.

Contribution

The paper presents a novel, simplified method based on regular bipartite graphs to unify and extend multiple classical theorems in intersecting families.

Findings

Unified proof technique for multiple theorems

New stronger results in intersecting families

Simplification of extremal set theory proofs

Abstract

In this paper we present a simple unifying approach to prove several statements about intersecting and cross-intersecting families, including the Erd\H os--Ko--Rado theorem, the Hilton--Milner theorem, a theorem due to Frankl concerning the size of intersecting families with bounded maximal degree, and versions of results on the sum of sizes of non-empty cross-intersecting families due to Frankl and Tokushige. Several new stronger results are also obtained. Our approach is based on the use of regular bipartite graphs. These graphs are quite often used in Extremal Set Theory problems, however, the approach we develop proves to be particularly fruitful.