On a $d$-degree Erdős-Ko-Rado Theorem

A family of subsets $\mathcal{F}$ is intersecting if $A\cap B\mathrm{\ne }\mathrm{\varnothing }$ for any $A,B\in \mathcal{F}$. In this paper, we show that for given integers $k>d\ge 2$ and $n\ge 2k+2d-3$, and any intersecting family $\mathcal{F}$ of $k$-subsets of $\{1,\cdots ,n\}$, there exists a $d$-subset of $[n]$ contained in at most $\left(\genfrac{}{}{0px}{}{n-d-1}{k-d-1}\right)$ subsets of $\mathcal{F}$. This result, proved using spectral graph theory, gives a $d$-degree generalization of the celebrated Erdős-Ko-Rado Theorem, improving a theorem of Kupavskii.