Using the existence of t-designs to prove Erd\H{o}s-Ko-Rado

In 1984, Wilson proved the Erd\H{o}s-Ko-Rado theorem for $t$-intersecting families of $k$-subsets of an $n$-set: he showed that if $n\ge (t+1)(k-t+1)$ and $\mathcal{F}$ is a family of $k$-subsets of an $n$-set such that any two members of $\mathcal{F}$ have at least $t$ elements in common, then $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\le \left(\genfrac{}{}{0px}{}{n-t}{k-t}\right)$. His proof made essential use of a matrix whose origin is not obvious. In this paper we show that this matrix can be derived, in a sense, as a projection of $t$-$(n,k,1)$ design.