Note on the codegree version of the Erd\H{o}s--Ko--Rado theorem

TL;DR

This paper improves bounds on the size of the ground set needed for certain intersecting families in combinatorics, specifically refining the codegree version of the Erd ext{o}s--Ko--Rado theorem for specific parameters.

Contribution

The authors provide tighter bounds on the size of the ground set for the codegree version of the Erd ext{o}s--Ko--Rado theorem when d=k-1 and d=k-2.

Findings

Bound on n for d=k-1 improved to 2k + √(2k) + O(1)

Bound on n for d=k-2 improved to 2k + 7k^{2/3} + O(k^{1/3})

Method extends previous results to new parameter ranges.

Abstract

Kupavskii proved a codegree version of the Erd\H{o}s--Ko--Rado theorem by showing that for an intersecting family $\mathcal{F} \subseteq \binom{[n]}{k}$ with $n \geq 2k +3d/(1-d/k)$, the minimum $d$-degree of $\mathcal{F}$ is at most $\binom{n-d-1}{k-d-1}$. Huang and Zhang improved the bound on $n$ to $n \geq 2k+2d-3$. In this short note, we prove that if $d = k-1$, then the bound on $n$ can be improved to $2k + \sqrt{2k} + O(1)$. In addition, we extend our method to show that the bound on $n$ can be improved to $2k + 7k^{2/3}+O(k^{1/3})$ when $d=k-2$.