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

Abstract: Let $\mathcal{F}\subset\binom{[n]}{k}$ 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 $d_{n}\le\binom{n-2}{k-2}$, 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 $d_{2k+1}\le\binom{n-2}{k-2}$, thereby confirming a conjecture of Frankl and Wang. Furthermore, we prove that for $k>50$ and $n>\frac{11}{2}k$, the $(k+2)$-th largest degree $d_{k+2}$ is already at most $\binom{n-2}{k-2}$. The techniques we developed also yield an tight upper bound for the $(\ell+1)$-th largest degree $d_{\ell+1}$ for $\ell \le k$ and sufficiently large $n$.