Multicolored lines in a finite geometry

Z. Füredi, I. G. Rosenberg

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Let t>1, let P1,...,Pt be pairwise disjoint nonvoid subsets of a finite set P, and let L be a collection of subsets of P called lines. We say that (P1,...,Pt;L) is a colored incidence structure if (i) each line meets at least two blocks Pi and Pj, (ii) for arbitrary distinct xε{lunate}Pi and yε{lunate}Pj the pair {x, y} is a subset of exactly one line if i≠j and at most one line if i=j. Extending the work of de Bruijn and Erdös and Meshulam we show that for a coloured incidence structure (P1,...,Pt;L) with |P1|≤...≤|Pt| in general |L| exceeds 1+|P1|+...+ |Pt-1|. The exceptional cases are exactly: (i) |L|=|P1|+·+|Pt-1| if the structur e is a truncated projective plane and (ii) |L|=1+|P1|+·+{divides}Pt-1| holds in exactly six cases: a projective plane, the dual of an affine plane, the dual of a modification of an affine plane, a near-pencil, a structure with 2 lines and a 9-element structure with 7 lines.

Original languageEnglish
Pages (from-to)149-163
Number of pages15
JournalDiscrete Mathematics
Volume71
Issue number2
DOIs
Publication statusPublished - 1988

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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