A Common Generalization of the Theorems of Erdős-Ko-Rado and Hilton-Milner

Abstract

Let $m$, $n$, and $k$ be integers satisfying $0<k\le n<2k\le m$. A family of sets $\mathcal{F}$ is called an $(m,n,k)$-intersecting family if $\left(\genfrac{}{}{0ex}{}{[n]}{k}\right)\subseteq \mathcal{F}\subseteq \left(\genfrac{}{}{0ex}{}{[m]}{k}\right)$ and any pair of members of $\mathcal{F}$ have nonempty intersection. Maximum $(m,k,k)$- and $(m,k+1,k)$-intersecting families are determined by the theorems of Erdős-Ko-Rado and Hilton-Milner, respectively. We determine the maximum families for the cases $n=2k-1,2k-2,2k-3$, and $m$ sufficiently large.

16 pages