A Common Generalization of the Theorems of Erd\H{o}s-Ko-Rado and Hilton-Milner

TL;DR

This paper unifies and extends classical theorems by Erd ext{"o}s-Ko-Rado and Hilton-Milner, determining maximum intersecting families for specific parameter ranges and large $m$, broadening understanding of set intersection properties.

Contribution

It generalizes the maximum size results of intersecting families for new parameter cases, unifying previous theorems under a common framework.

Findings

Maximum families for $n=2k-1$

Maximum families for $n=2k-2$

Maximum families for $n=2k-3$

Abstract

Let $m$, $n$, and $k$ be integers satisfying $0 < k \leq n < 2k \leq m$. A family of sets $\mathcal{F}$ is called an $(m,n,k)$-intersecting family if $\binom{[n]}{k} \subseteq \mathcal{F} \subseteq \binom{[m]}{k}$ 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 Erd\H{o}s-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.