DMTCS Proceedings, 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

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Color critical hypergraphs and forbidden configurations

Richard Anstee, Balin Fleming, Zoltán Füredi, Attila Sali


The present paper connects sharpenings of Sauer's bound on forbidden configurations with color critical hypergraphs. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. Let F be a k× l (0,1)-matrix (the forbidden configuration). Assume A is an m× n simple matrix which has no submatrix which is a row and column permutation of F. We define forb(m,F) as the best possible upper bound on n, for such a matrix A, which depends on m and F. It is known that forb(m,F)=O(mk) for any F, and Sauer's bond states that forb(m,F)=O(mk-1) fore simple F. We give sufficient condition for non-simple F to have the same bound using linear algebra methods to prove a generalization of a result of Lovász on color critical hypergraphs.

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