A short proof of a cross-intersection theorem of Hilton

Families A1,…,AkA1,…,Ak of sets are said to be cross-intersecting   if Ai∩Aj≠0̸ for any Ai∈AiAi∈Ai and Aj∈AjAj∈Aj, i≠ji≠j. A nice result of Hilton that generalises the Erdős–Ko–Rado (EKR) Theorem says that if r≤n/2r≤n/2 and A1,…,AkA1,…,Ak are cross-intersecting sub-families of [n]r, then ∑i=1k|Ai|≤{nrif k≤nr;kn−1r−1if k≥nr, and the bounds are best possible. We give a short proof of a slightly stronger version. For this purpose, we extend Daykin’s proof of the EKR Theorem to obtain the following improvement of the EKR Theorem: if r≤n/2r≤n/2, A⊆[n]r, A∗≔{A∗∈A:A∗∩A≠0̸ for all A∈A}A∗≔{A∗∈A:A∗∩A≠0̸ for all A∈A} and A′≔A∖A∗A′≔A∖A∗, then |A∗|+rn|A′|≤n−1r−1.