Classification of Maximum Hittings by Large Families

For integers r and n , where n is sufficiently large, and for every set X ⊆ [n] we determine the maximal left-compressed intersecting families 𝒜⊆( [ [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.