Short injective proofs of the Erd\H{o}s-Ko-Rado and Hilton-Milner Theorem: A canonical partition of shifted intersecting set systems

TL;DR

This paper introduces a canonical partition of shifted intersecting set systems, enabling simplified proofs of key combinatorial theorems and characterizing maximal systems, advancing understanding in extremal set theory.

Contribution

It provides a unified framework for proofs of the Erd ext{o}s-Ko-Rado and Hilton-Milner Theorems through a canonical partition approach.

Findings

Unified proofs of Erd ext{o}s-Ko-Rado and Hilton-Milner Theorems

Characterization of maximal shifted k-uniform intersecting systems

Canonical partition method for shifted intersecting set systems

Abstract

We give a canonical partition of shifted intersecting set systems, from which one can obtain unified and elementary proofs of the Erd\H{o}s-Ko-Rado and Hilton-Milner Theorem, as well as a characterization of maximal shifted $k$-uniform intersecting set systems over $[n]$.