On the nonuniform Fisher inequality
Discrete Mathematics
Geometric matrices and an inequality for (0,1)-matrices
Discrete Mathematics
A sharpening of Fisher's inequality
Discrete Mathematics
On generalizations of the deBruijn-Erdo¨s theorem
Journal of Combinatorial Theory Series A
Proof of a conjecture of Frankl and Füredi
Journal of Combinatorial Theory Series A
Sylvester–Gallai Theorem and Metric Betweenness
Discrete & Computational Geometry
Problems related to a de Bruijn-Erdös theorem
Discrete Applied Mathematics
Graph Theory
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One of the De Bruijn-Erd驴s theorems deals with finite hypergraphs where every two vertices belong to precisely one hyperedge. It asserts that, except in the perverse case where a single hyperedge equals the whole vertex set, the number of hyperedges is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of simply described families, near-pencils and finite projective planes. Chen and Chvátal proposed to define the line uv in a 3-uniform hypergraph as the set of vertices that consists of u, v, and all w such that {u;v;w} is a hyperedge. With this definition, the De Bruijn-Erd驴s theorem is easily seen to be equivalent to the following statement: If no four vertices in a 3-uniform hypergraph carry two or three hyperedges, then, except in the perverse case where one of the lines equals the whole vertex set, the number of lines is at least the number of vertices and the two numbers are equal if and only if the hypergraph belongs to one of two simply described families. Our main result generalizes this statement by allowing any four vertices to carry three hyperedges (but keeping two forbidden): the conclusion remains the same except that a third simply described family, complements of Steiner triple systems, appears in the extremal case.