Finite linear spaces and projective planes

In 1948, De Bruijn and Erdös proved that a finite linear space on v points has at least v lines, with equality occurring if and only if the space is either a near-pencil (all points but one collinear) or a projective plane . In this paper, we study finite linear spaces which are not near-pencils . We obtain a lower bound for the number of lines (as a function of the number of points) for such l...

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Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 2

The flag geometry 1=(P, L, I) of a finite projective plane 6 of order s is the generalized hexagon of order (s, 1) obtained from 6 by putting P equal to the set of all flags of 6, by putting L equal to the set of all points and lines of 6, and where I is the natural incidence relation (inverse containment), i.e., 1 is the dual of the double of 6 in the sense of H. Van Maldeghem (1998, ``General...

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Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 1

The flag geometry 1=(P, L, I) of a finite projective plane 6 of order s is the generalized hexagon of order (s, 1) obtained from 6 by putting P equal to the set of all flags of 6, by putting L equal to the set of all points and lines of 6, and where I is the natural incidence relation (inverse containment), i.e., 1 is the dual of the double of 6 in the sense of H. Van Maldeghem (1998, ``General...

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Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 3

The flag geometry 1=(P, L, I) of a finite projective plane 6 of order s is the generalized hexagon of order (s, 1) obtained from 6 by putting P equal to the set of all flags of 6, by putting L equal to the set of all points and lines of 6, and where I is the natural incidence relation (inverse containment), i.e., 1 is the dual of the double of 6 in the sense of H. Van Maldeghem (1998, ``General...

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On Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces

The flag geometry = (P,L, I) of a finite projective plane ⇧ of order s is the generalized hexagon of order (s, 1) obtained from ⇧ by putting P equal to the set of all flags of ⇧, by putting L equal to the set of all points and lines of ⇧ and where I is the natural incidence relation (inverse containment), i.e., is the dual of the double of ⇧ in the sense of [8]. Then we say that is fully (and w...

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Finite linear spaces and projective planes

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