On a $d$-degree Erd\H{o}s-Ko-Rado Theorem

TL;DR

This paper generalizes the Erd ext{"o}s-Ko-Rado Theorem to $d$-degree settings using spectral graph theory, providing bounds on the number of $k$-subsets containing a fixed $d$-subset in intersecting families.

Contribution

It introduces a $d$-degree generalization of the Erd ext{"o}s-Ko-Rado Theorem, improving previous bounds by Kupavskii through spectral graph theory techniques.

Findings

Establishes bounds on $d$-subsets in intersecting families.

Uses spectral graph theory for combinatorial bounds.

Generalizes classical Erd ext{"o}s-Ko-Rado results.

Abstract

A family of subsets $\mathcal{F}$ is intersecting if $A \cap B \neq \emptyset$ 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 $\binom{n-d-1}{k-d-1}$ subsets of $\mathcal{F}$. This result, proved using spectral graph theory, gives a $d$-degree generalization of the celebrated Erd\H{o}s-Ko-Rado Theorem, improving a theorem of Kupavskii.