New injective proofs of the Erdős-Ko-Rado and Hilton-Milner theorems

A set system F is intersecting if for any F,F′∈F, F∩F′≠∅. A fundamental theorem of Erdős, Ko and Rado states that if F is an intersecting family of r-subsets of [n]={1,…,n}, and n≥2r, then |F|≤n−1r−1. Furthermore, when n>2r, equality holds if and only if F is the family of all r-subsets of [n] containing a fixed element. This was proved as part of a stronger result by Hilton and Milner. In this note, we provide new injective proofs of the Erdős-Ko-Rado and the Hilton-Milner theorems.