Improved bounds on the maximum diversity of intersecting families

Abstract

A family F⊂[n]k is called an intersecting family if F∩F′≠0̸ for all F,F′∈F. If ∩F≠0̸ then F is called a star. The diversity of an intersecting family F is defined as the minimum number of k-sets in F, whose deletion results in a star. In the present paper, we prove that for n>36k any intersecting family F⊂[n]k has diversity at most n−3k−2, which improves the previous best bound n>72k due to the first author. This result is derived from some strong bounds concerning the maximum degree of large intersecting families. Some related results are established as well.

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