A remark on the $t$-intersecting Erd\H{o}s-Ko-Rado theorem

TL;DR

This paper demonstrates the equivalence of two linear algebraic proofs of the $t$-intersecting Erd ext{o}s-Ko-Rado theorem, unifying different approaches in combinatorics.

Contribution

It shows that Schrijver's and Wilson's linear algebraic proofs of the theorem are fundamentally equivalent, clarifying the relationship between these methods.

Findings

Proves the equivalence of two linear algebraic proofs

Unifies different proof techniques for the theorem

Clarifies the theoretical understanding of the proof methods

Abstract

The $t$-intersecting Erd\H{o}s-Ko-Rado theorem is the following statement: if $\mathcal{F} \subset \binom{[n]}{k}$ is a $t$-intersecting family of sets and $n\ge (t+1)(k-t+1)$, then $|\mathcal{F}| \le \binom{n-t}{k-t}$. The first proof of this statement for all $t$ was a linear algebraic argument of Wilson. Earlier, Schrijver had proven the $t$-intersecting Erd\H{o}s-Ko-Rado theorem for sufficiently large $n$ by a seemingly different linear algebraic argument motivated by Delsarte theory. In this note, we show that the approaches of Schrijver and Wilson are in fact equivalent.