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

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.