On the largest degrees in intersecting hypergraphs

TL;DR

This paper investigates the maximum degrees in intersecting hypergraphs, improving bounds on the second largest degree and establishing optimal bounds for the third and other degrees, advancing understanding of hypergraph degree distributions.

Contribution

It strengthens existing bounds on the second largest degree and provides new optimal bounds for the third and other degrees in intersecting hypergraphs for large n.

Findings

The second largest degree is at most rom inom{n-2}{k-2} for n rom 6k-9.

The third largest degree is at most rom inom{n-2}{k-2}+inom{n-3}{k-2}.

Several bounds of similar nature are proven to be best possible.

Abstract

Let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-set $[n]=\{1,2,\ldots,n\}$. Let $n>2k$ and let $\mathcal{F}\subset \binom{[n]}{k}$ be an {\it intersecting} $k$-graph, i.e., $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. The number of edges $F\in \mathcal{F}$ containing $x\in [n]$ is called the {\it degree} of $x$. Assume that $d_1\geq d_2\geq \ldots\geq d_n$ are the degrees of $\mathcal{F}$ in decreasing order. An important result of Huang and Zhao states that for $n>2k$ the minimum degree $d_n$ is at most $\binom{n-2}{k-2}$. For $n\geq 6k-9$ we strengthen this result by showing $d_{2k+1}\leq \binom{n-2}{k-2}$. As to the second and third largest degrees we prove the best possible bound $d_3\leq d_2\leq \binom{n-2}{k-2}+\binom{n-3}{k-2}$ for $n>2k$. Several more best possible results of a similar nature are established.