Intersecting kk-uniform families containing a given family

A family AA of sets is said to be intersecting if A∩B≠∅A∩B≠∅ for any A,B∈AA,B∈A. Let m,n,km,n,k and rr be positive integers with m≥2k≥2r>n≥km≥2k≥2r>n≥k. A family FF of sets is called an (m,n,k,r)(m,n,k,r)-intersecting family   if FF is an intersecting subfamily of [m]k containing {A∈[m]k:|A∩[n]|≥r}. Maximum (m,k,k,k)(m,k,k,k)- and (m,k+1,k,k)(m,k+1,k,k)-intersecting families were determined by the well-known Erdős–Ko–Rado theorem and Hilton–Milner theorem, respectively. Recently, Li et al. determined maximum (m,n,k,k)(m,n,k,k)-intersecting family when n=2k−1,2k−2,2k−3n=2k−1,2k−2,2k−3 or mm is sufficiently large. In this paper, we determine all the maximum (m,n,k,r)(m,n,k,r)-intersecting families.