Multicolored lines in a finite geometry
type: Item
instance of: scholarly article
title: Multicolored lines in a finite geometry
zbMATH Open document ID: 0661.05021
DOI: 10.1016/0012-365X(88)90068-4
published in: Discrete Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let P be a set of points, partitioned into blocks \(P_ 1,...,P_ t\), and let \underbar{L} be a set of lines (subsets of P) satisfying the following conditions: (i) each line intersects at least two of the blocks (ii) any two points in distinct blocks are on a unique line (iii) any two points in the same block are on at most one line. Then \((P_ 1,...,P_ t;\underline{L})\) is called colored incidence structure. (One may think of this as the structure induced by a (partial) coloration of a finite linear space, where one discards colorless and monochromatic lines.) Extending a result of \textit{R. Meshulam} [``On multicolored lines in a linear space'', J. Comb. Theory, Ser. A 40, 150-155 (1985; Zbl 0616.05003)] the authors show that one always has \(| \underline{L}| \geq | P_ 1| +...+| P_{t-1}|\) and that in general even \(| \underline{L}| >| P_ 1| +...+| P_{t-1}|.\) They completely classify the cases where \(| L| =(1+)| P_ 1| +...+| P_{t-1}|;\) one obtains (truncated) projective planes, (modified) dual affine planes, near-pencils - a structure with 2 lines and a structure on 9 points with 7 lines.
reviewed by: Dieter Jungnickel
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51E30
zbMATH DE Number: 4079444
zbMATH Keywords: finite geometries
zbMATH Keywords: colored incidence structure
zbMATH Keywords: linear space
author: Zoltán Füredi
author: Ivo G. Rosenberg
MaRDI profile type: Publication
cites work: Q5790167
cites work: Q5678865
cites work: An improved bound for extending partial projective planes
cites work: Finite linear spaces and projective planes
cites work: On multi-coloured lines in a linear space
cites work: Packing nearly-disjoint sets
cites work: Q3333054
full work available at URL: 0012-365x(88)90068-4
OpenAlex ID: W2022037033
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description: scientific article; zbMATH DE number 4079444