On the largest degrees in intersecting hypergraphs
Peter Frankl, Jian Wang
TL;DR
This work investigates the largest degrees in intersecting $k$-graphs on $[n]$, aiming to sharpen Huang–Zhao’s bound by analyzing not only the minimum degree but also the second, third, and higher largest degrees. The authors identify the triangle (or $\mathcal{H}_2$) family as extremal for $d_2$ and $d_3$, establishing sharp upper bounds and equality cases, and they prove $d_{2k+1}(\mathcal{F})\le\binom{n-2}{k-2}$ for $n\ge6k-9$, with further high-degree bounds like $d_{\lceil 8k/3\rceil}(\mathcal{F})\le\binom{n-2}{k-2}$ in broader regimes. They extend the analysis to $t$-intersecting families, deriving improved bounds for larger $n$ through a combination of shifting, Kruskal–Katona theory, sunflower arguments, and Daykin-type cross-intersection results, including a result that bounds $d_{1+\tau_t(\mathcal{F})}$ when $\tau_t(\mathcal{F})\ge t+2$ and structural results for $\tau_t(\mathcal{F})\le t+1$. The paper closes with conjectures on extending these bounds to wider ranges of $n$ and $k$, highlighting the transversal- and basis-related structure as a unifying framework for high-degree bounds in intersecting hypergraphs.
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.