On the largest degrees in intersecting hypergraphs

Peter Frankl; Jian Wang

paper

On the largest degrees in intersecting hypergraphs

arXiv:2511.15508

Abstract

Let $(k [ n ])$ denote the collection of all $k$ -subsets of the standard $n$ -set $[n] = {1, 2, \dots, n}$ . Let $n > 2 k$ and let $F \subset (k [ n ])$ be an {\it intersecting} $k$ -graph, i.e., $F \cap F^{'} \neq = \emptyset$ for all $F, F^{'} \in F$ . The number of edges $F \in F$ containing $x \in [n]$ is called the {\it degree} of $x$ . Assume that $d_{1} \geq d_{2} \geq \dots \geq d_{n}$ are the degrees of $F$ in decreasing order. An important result of Huang and Zhao states that for $n > 2 k$ the minimum degree $d_{n}$ is at most $(k - 2 n - 2)$ . For $n \geq 6 k - 9$ we strengthen this result by showing $d_{2 k + 1} \leq (k - 2 n - 2)$ . As to the second and third largest degrees we prove the best possible bound $d_{3} \leq d_{2} \leq (k - 2 n - 2) + (k - 2 n - 3)$ for $n > 2 k$ . Several more best possible results of a similar nature are established.