Simple Algebraic Proofs of Uniqueness for Erd\H{o}s-Ko-Rado Theorems

TL;DR

This paper presents simplified algebraic proofs confirming the uniqueness of largest intersecting families in Erdős-Ko-Rado theorems and characterizes maximal partially 2-intersecting families of perfect hypermatchings, resolving a recent conjecture.

Contribution

It introduces simpler algebraic methods for proving uniqueness in Erdős-Ko-Rado theorems and characterizes maximal partially 2-intersecting families, solving a recent conjecture.

Findings

Simpler algebraic proofs of uniqueness for Erdős-Ko-Rado theorems

Characterization of largest partially 2-intersecting families of perfect hypermatchings

Resolution of a recent conjecture by Meagher, Shirazi, and Stevens

Abstract

We give simpler algebraic proofs of uniqueness for several Erd\H{o}s-Ko-Rado results, i.e., that the canonically intersecting families are the only largest intersecting families. Using these techniques, we characterize the largest partially 2-intersecting families of perfect hypermatchings, resolving a recent conjecture of Meagher, Shirazi, and Stevens.