# A degree version of the Hilton--Milner theorem

## TL;DR

This paper extends the Hilton--Milner theorem by establishing a degree version for non-trivial intersecting families of sets, showing that the minimum degree is maximized by a specific extremal family.

## Contribution

It introduces a degree-based stability result for intersecting families, generalizing the classical Hilton--Milner theorem with a novel proof approach.

## Key findings

- Minimum degree of non-trivial intersecting families is bounded by that of a specific extremal family.
- The degree version holds when n=Ω(k^2), extending previous size-based results.
- Uses fundamental extremal set theory results and introduces a new variant of the Erd ext{"o}s--Ko--Rado theorem.

## Abstract

An intersecting family of sets is trivial if all of its members share a common element. Hilton and Milner proved a strong stability result for the celebrated Erd\H{o}s--Ko--Rado theorem: when $n> 2k$, every non-trivial intersecting family of $k$-subsets of $[n]$ has at most $\binom{n-1}{k-1}-\binom{n-k-1}{k-1}+1$ members. One extremal family $\mathcal{HM}_{n, k}$ consists of a $k$-set $S$ and all $k$-subsets of $[n]$ containing a fixed element $x\not\in S$ and at least one element of $S$. We prove a degree version of the Hilton--Milner theorem: if $n=\Omega(k^2)$ and $\mathcal{F}$ is a non-trivial intersecting family of $k$-subsets of $[n]$, then $\delta(\mathcal{F})\le \delta(\mathcal{HM}_{n.k})$, where $\delta(\mathcal{F})$ denotes the minimum (vertex) degree of $\mathcal{F}$. Our proof uses several fundamental results in extremal set theory, the concept of kernels, and a new variant of the Erd\H{o}s--Ko--Rado theorem.