Improved Bounds Concerning the Maximum Degree of Intersecting Hypergraphs

For positive integers $n>k>t$ let [n]k denote the collection of all k-subsets of the standard n-element set [n]=\1,…,n\. Subsets of [n]k are called k-graphs. A k-graph F is called t-intersecting if |F∩ F'|≥ t for all F,F'∈ F. One of the central results of extremal set theory is the Erdős-Ko-Rado Theorem which states that for n≥ (k-t+1)(t+1) no t-intersecting k-graph has more than n-tk-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 F(i)= \ i∈ F∈ F\. The quantity (F)=max1≤ i≤ n|F(i)|/|F| measures how close a k-graph is to a star. The main result (Theorem 1.3) shows that (F)>1/d holds if F is 1-intersecting, |F|>2ᵈd²ᵈ⁺¹n-d-1k-d-1 and n≥ 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≥ 2 (Theorem 1.4).