On the $\ell$-th largest degree of an intersecting family

Abstract

Let $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ be an intersecting family. For an element $i\in [n]$, the degree of $i$ is the number of sets in $\mathcal{F}$ that contain $i$. Assume that the degrees are ordered as $d_1\ge d_2\ge \cdots \ge d_n$.Huang and Zhao showed that if $n>2k$, then the minimum degree satisfies $d_n\le \left(\genfrac{}{}{0ex}{}{n-2}{k-2}\right)$, with the maximum attained by the $1$-star. We strengthen this result by proving that for $n\ge 2k+1$, the $(2k+1)$-th largest degree satisfies $d_{2k+1}\le \left(\genfrac{}{}{0ex}{}{n-2}{k-2}\right)$, thereby confirming a conjecture of Frankl and Wang. Furthermore, we prove that for large $k$ and $n>12k$, the $(k+2)$-th largest degree $d_{k+2}$ is already at most $\left(\genfrac{}{}{0ex}{}{n-2}{k-2}\right)$. The techniques we developed also yield a tight upper bound for the $(\ell +1)$-th largest degree $d_{\ell +1}$ for $\epsilon k\le \ell \le k$ and sufficiently large $n>C_{\epsilon }k$.