Classification of maximum hittings by large families

TL;DR

This paper characterizes the largest left-compressed intersecting families in combinatorics that maximize intersection with a given set, extending prior work and answering a specific open question.

Contribution

It determines the maximal left-compressed intersecting families that maximize hitting with a set X, generalizing previous results and answering Barber's question.

Findings

Identifies families achieving maximum hitting for large n

Extends Borg's results to broader classes of sets X

Provides a complete characterization of optimal families

Abstract

For integers $r$ and $n$, where $n$ is sufficiently large, and for every set $X \subseteq [n]$ we determine the maximal left-compressed intersecting families $\mathcal{A}\subseteq \binom{[n]}{r}$ which achieve maximum hitting with $X$ (i.e. have the most members which intersect $X$). This answers a question of Barber, who extended previous results by Borg to characterise those sets $X$ for which maximum hitting is achieved by the star.