Maximum hitting for n sufficiently large

TL;DR

This paper characterizes, for large n, the sets X for which the maximum size of intersecting families containing X is achieved by the star family, answering a question in extremal combinatorics.

Contribution

It provides a complete characterization of the sets X that maximize the size of intersecting families containing X for large n, extending previous partial results.

Findings

Identifies exactly which sets X maximize |A(X)| for large n.

Shows that the maximum is achieved by the star family for these X.

Provides a precise threshold for n depending on r.

Abstract

For a left-compressed intersecting family \A contained in [n]^(r) and a set X contained in [n], let \A(X) = {A in \A : A intersect X is non-empty}. Borg asked: for which X is |\A(X)| maximised by taking \A to be all r-sets containing the element 1? We determine exactly which X have this property, for n sufficiently large depending on r.