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

TL;DR

This paper demonstrates that Wilson's proof of the Erd ext{"o}s-Ko-Rado theorem can be understood through the lens of $t$-designs, providing a new perspective on the matrix used in the original proof.

Contribution

The paper reveals that Wilson's key matrix in the Erd ext{"o}s-Ko-Rado theorem proof can be derived from $t$-$(n,k,1)$ designs, offering a novel conceptual approach.

Findings

Matrix can be viewed as a projection of a $t$-$(n,k,1)$ design.

Provides a new proof perspective for Erd ext{"o}s-Ko-Rado theorem.

Connects combinatorial designs with classical intersection theorems.

Abstract

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 $|\mathcal{F}|\le\binom{n-t}{k-t}$. 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.