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Maximal intersection critical families of finite sets

✍ Scribed by N.Zagaglia Salvi

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 129 KB
Volume: 155
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(94)00391-u

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✦ Synopsis

A finite family of pairwise intersecting r-sets is a maximal r-clique if it cannot be extended to another r-clique by adding a new r-set. It is intersection critical if it is not possible to replace any edge by some of its proper subsets, without violating the intersection property.

We prove that if a maximal r-clique H, distinct from K;+ 1, is not intersection critical, then IHI > I V(H)I.

Moreover, we prove that the system of lines of a projective plane not passing through a fixed point is an intersection critical r-clique, not contained in any larger one.

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