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How many lines must miss d points in a linear space
Author(s) -
Metsch Klaus
Publication year - 2006
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/jcd.20116
Subject(s) - mathematics , combinatorics , linear space , space (punctuation) , projective plane , projective space , plane (geometry) , algebraic number , degenerate energy levels , discrete mathematics , point (geometry) , geometry , pure mathematics , projective test , mathematical analysis , philosophy , linguistics , physics , quantum mechanics , correlation
It was remarked by P. Erdős, J. C. Fowler, V. T. Sós, and R. M. Wilson, (J Combin Theory Ser A 38, 1985, 131–142), that a non‐degenerate finite linear space on v points has the following property. For every point P the number of lines not passing through P is at least $\lfloor v-\sqrt v\rfloor$ with equality if and only if the linear space is a projective plane. We give a short proof for this result using an algebraic tool. The main purpose of the article however is to generalize this result in the direction that gives a lower bound for the number of lines not passing through a given number d of points. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 441–450, 2006