Elsevier

Discrete Mathematics

Volume 155, Issues 1–3, 1 August 1996, Pages 267-269
Discrete Mathematics

Maximal intersection critical families of finite sets

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Abstract

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 Kr+1r is not intersection critical, then | H | > | V (H) |.

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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Research supported by M.U.R.S.T. (Ministero dell'Universita' e della Ricerca Scientifica e Tecnologica).