New injective proofs of the Erd\H{o}s--Ko--Rado and Hilton--Milner theorems

TL;DR

This paper presents new injective proofs for the Erd ext{"o}s--Ko--Rado and Hilton--Milner theorems, which characterize the maximum size of intersecting set families of subsets.

Contribution

The authors introduce novel injective proof techniques for fundamental theorems in extremal set theory, providing alternative approaches to classical results.

Findings

New injective proofs for Erd ext{"o}s--Ko--Rado theorem

New injective proofs for Hilton--Milner theorem

Enhanced understanding of intersecting set families

Abstract

A set system F is intersecting if any pair of sets in F have a nonempty intersection. A fundamental theorem of Erd\H{o}s, Ko and Rado states that if F is an intersecting family of r-subsets of [n]={1,...,n}, and n>= 2r, then the cardinality of F is at most the cardinality of the family of all r-subsets of [n] containing a fixed element. Furthermore, when n>2r, equality holds if and only if F is the family of all r-subsets of [n] containing a fixed element. This characterization was proved as part of a stronger result by Hilton and Milner. In this note, we provide new injective proofs of the Erd\H{o}s--Ko--Rado and the Hilton--Milner theorems.