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A set system F is intersecting if any pair of sets in F have a nonemptyintersection. A fundamental theorem of Erd\H{o}s, Ko and Rado states that if Fis an intersecting family of r-subsets of [n]={1,...,n}, and n>= 2r, then thecardinality of F is at most the cardinality of the family of all r-subsets of[n] containing a fixed element. Furthermore, when n>2r, equality holds if andonly if F is the family of all r-subsets of [n] containing a fixed element.This characterization was proved as part of a stronger result by Hilton andMilner. In this note, we provide new injective proofs of theErd\H{o}s--Ko--Rado and the Hilton--Milner theorems.

New injective proofs of the Erd\H{o}s--Ko--Rado and Hilton--Milner theorems