Improved bounds on the maximum diversity of intersecting families

A family F subset of (([n])(k)) is called an intersecting family if F boolean AND F not equal empty set for all F, F' is an element of F. If boolean AND F not equal empty set 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 subset of (((k))([n])) has diversity at most ((n-3)(k-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. (c) 2023 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Automated summary

In this paper, it is proven that for n > 36k, any intersecting family F subset of (((k))([n])) has a diversity of at most ((n-3)(k-2)), improving upon the previous best bound n > 72k.