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A characterization of projective 3‐spaces
Author(s) -
Durante N.,
Metsch K.
Publication year - 1996
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/(sici)1520-6610(1996)4:5<377::aid-jcd6>3.0.co;2-c
Subject(s) - mathematics , projective plane , projective space , combinatorics , real projective plane , space (punctuation) , plane (geometry) , planar , de bruijn sequence , prime (order theory) , characterization (materials science) , projective test , complex projective space , pure mathematics , geometry , physics , linguistics , philosophy , computer graphics (images) , computer science , correlation , optics
It is known that if L is a nondegenerate linear space with v points and if P is a point of L, there exist at least [ v − √ v ] lines that do not contain P with equality iff L is a projective plane. This result is stronger than the famous de Bruijn‐Erdös Theorem, which states that every nondegenerate linear space has at least as many lines as points with equality iff it is a projective plane. We prove the following analogous theorem for planar spaces. Suppose that S is a planar space with p 3 + p 2 + p + 1 points for a real number p > 1. If P is a point of S, then there exist at least p 3 planes that do not contain P with equality if and only if p is a prime power and S is the projective 3‐space PG (3, p ). © 1996 John Wiley & Sons, Inc.