Note on the codegree version of the Erdős--Ko--Rado theorem

Abstract

Kupavskii proved a codegree version of the Erdős--Ko--Rado theorem by showing that for an intersecting family $\mathcal{F}\subseteq \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ with $n\ge 2k+3d/(1-d/k)$, the minimum $d$-degree of $\mathcal{F}$ is at most $\left(\genfrac{}{}{0ex}{}{n-d-1}{k-d-1}\right)$. Huang and Zhang improved the bound on $n$ to $n\ge 2k+2d-3$. 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 $d=k-2$.

10 pages