An algebraic groups perspective on Erd\H{o}s-Ko-Rado

TL;DR

This paper offers an algebraic group theory proof of the Erdős-Ko-Rado Theorem, establishing a novel connection with the Gerstenhaber Theorem on nilpotent matrices, and providing new insights into combinatorial and algebraic structures.

Contribution

It introduces an algebraic groups perspective to prove the Erdős-Ko-Rado Theorem, linking combinatorics with algebraic geometry and group theory.

Findings

Proof using Borel Fixed Point Theorem

Strong analogy with Gerstenhaber Theorem

New algebraic perspective on combinatorial theorems

Abstract

We give a proof of the Erd\H{o}s-Ko-Rado Theorem using the Borel Fixed Point Theorem from algebraic group theory. This perspective gives a strong analogy between the Erd\H{o}s-Ko-Rado Theorem and (generalizations of) the Gerstenhaber Theorem on spaces of nilpotent matrices.