## A simple proof of the Hilton-Milner theorem.(English)Zbl 1414.05289

Summary: Let $$n \geq 2k \geq 4$$ be integers and $$\mathcal{F}$$ a family of $$k$$-subsets of $$\{1,2,\dots, n\}$$. We call $$\mathcal{F}$$ intersecting if $$F \cap F' \neq \emptyset$$ for all $$F, F' \in \mathcal{F}$$, and we call $$\mathcal{F}$$ nontrivial if $$\bigcap_{F \in \mathcal{F}} F = \emptyset$$. Strengthening the famous Erdős-Ko-Rado theorem, A. Hilton and E. Milner [Q. J. Math. Oxf. 18, 369–384 (1967; Zbl 0168.26205)] proved that $$|\mathcal{F}| \leq \smash{\binom{n - 1}{k - 1}} - \smash{\binom{n - k - 1}{k - 1}} + 1$$ if $$\mathcal{F}$$ is nontrivial and intersecting. We provide a proof by injection of this result.

### MSC:

 05D05 Extremal set theory

### Keywords:

finite sets; intersection; hypergraphs

Zbl 0168.26205
Full Text:

### References:

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