Improved Bounds Concerning the Maximum Degree of Intersecting Hypergraphs

Abstract

For positive integers $n>k>t$ let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-element set $[n]=\{1,\ldots,n\}$. Subsets of $\binom{[n]}{k}$ are called $k$-graphs. A $k$-graph $\mathcal{F}$ is called $t$-intersecting if $|F\cap F'|\geq t$ for all $F,F'\in \mathcal{F}$. One of the central results of extremal set theory is the Erdős-Ko-Rado Theorem which states that for $n\geq (k-t+1)(t+1)$ no $t$-intersecting $k$-graph has more than $\binom{n-t}{k-t}$ edges. For $n$ greater than this threshold the $t$-star (all $k$-sets containing a fixed $t$-set) is the only family attaining this bound. Define $\mathcal{F}(i)=\{F\setminus \{i\}\colon i\in F\in \mathcal{F}\}$. The quantity $\varrho(\mathcal{F})=\max\limits_{1\leq i\leq n}|\mathcal{F}(i)|/|\mathcal{F}|$ measures how close a $k$-graph is to a star. The main result (Theorem 1.3) shows that $\varrho(\mathcal{F})>1/d$ holds if $\mathcal{F}$ is 1-intersecting, $|\mathcal{F}|>2^dd^{2d+1}\binom{n-d-1}{k-d-1}$ and $n\geq 4(d-1)dk$. Such a statement can be deduced from earlier results, however only for much larger values of $n/k$ and/or $n$. The proof is purely combinatorial, it is based on a new method: shifting ad extremis. The same method is applied to obtain a nearly optimal bound in the case of $t\geq 2$ (Theorem 1.4).