A new short proof of the EKR theorem

TL;DR

This paper presents a new, shorter proof of the Erdős–Ko–Rado theorem, which bounds the size of intersecting families of k-subsets, improving on classical proofs in simplicity and elegance.

Contribution

The paper introduces a novel, more concise proof of the EKR theorem, simplifying the understanding of the combinatorial bound.

Findings

Proof is shorter than classical proofs by Katona and Daykin.

Confirms the maximum size of intersecting families of k-subsets.

Provides a more elegant combinatorial argument.

Abstract

A family F is intersecting if any two members have a nonempty intersection. Erdos, Ko, and Rado showed that |F|\leq {n-1\choose k-1} holds for an intersecting family of k-subsets of [n]:={1,2,3,...,n}, n\geq 2k. For n> 2k the only extremal family consists of all k-subsets containing a fixed element. Here a new proof is presented. It is even shorter than the classical proof of Katona using cyclic permutations, or the one found by Daykin applying the Kruskal-Katona theorem.