Maximum degree and diversity in intersecting hypergraphs

Let S be an n-element set and F subset of ((S)(k)) an intersecting family. Improving earlier results it is proved that for n > 72k there is an element of S that is contained in all but ((n-3)(k-2)) members of F. One of the main ingredients of the proof is the following statement. If G subset of ((S)(k)) is intersecting, vertical bar G vertical bar >= ((n-2)(k-2)) and n >= 72k then there is an element of S that is contained in more than half of the members of G. (C) 2020 Elsevier Inc. All rights reserved.