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Combinatorica

, Volume 12, Issue 1, pp 39–44 | Cite as

Bounds for arrays of dots with distinct slopes or lengths

  • Paul Erdős
  • Ron Graham
  • Imre Z. Ruzsa
  • Herbert Taylor
Article

Abstract

Ann×m sonar sequence is a subset of then×m grid with exactly one point in each column, such that the\(\mathop 2\limits^m \) vectors determined by them are all distinct. We show that for fixedn the maximalm for which a sonar sequence exists satisfiesnCn11/20<m<n+4n2/3 for alln andm>n+c logn log logn for infinitely manyn.

Another problem concerns the maximal numberD of points that can be selected from then×m grid so that all the\(\mathop 2\limits^D \) vectors have slopes. We proven1/2Dn4/5

AMS subject classification code (1991)

05 C 

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Copyright information

© Akadémiai Kiadó 1992

Authors and Affiliations

  • Paul Erdős
    • 1
  • Ron Graham
    • 2
  • Imre Z. Ruzsa
    • 1
  • Herbert Taylor
    • 3
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary
  2. 2.AT & T Bell LaboratoriesMurray HillUSA
  3. 3.Communication Sciences Institute EEB 500University of Southern CaliforniaLos AngelesUSA

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