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arXiv · 1609.04714

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

Glenn Hurlbert·Vikram Kamat
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Abstract

A set system F is intersecting if any pair of sets in F have a nonempty intersection. A fundamental theorem of Erd\H{o}s, Ko and Rado states that if F is an intersecting family of r-subsets of [n]={1,...,n}, and n>= 2r, then the cardinality 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 and only 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 and Milner. In this note, we provide new injective proofs of the Erd\H{o}s--Ko--Rado and the Hilton--Milner theorems.

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Glenn Hurlbert, Vikram Kamat. 2016-09-15. New injective proofs of the Erd\H{o}s--Ko--Rado and Hilton--Milner theorems. https://arxiv.org/abs/1609.04714

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