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Bounds for arrays of dots with distinct slopes or lengths

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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 satisfiesn−Cn 11/20<m<n+4n 2/3 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 proven 1/2≪D≪n 4/5

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Authors and Affiliations

  1. Mathematical Institute of the Hungarian Academy of Sciences, Reáltanoda u. 13-15, 1053, Budapest, Hungary

    Paul Erdős & Imre Z. Ruzsa

  2. AT & T Bell Laboratories, 600 Mountain Ave., 07974, Murray Hill, NJ, USA

    Ron Graham

  3. Communication Sciences Institute EEB 500, University of Southern California, 90089-2565, Los Angeles, CA, USA

    Herbert Taylor

Authors
  1. Paul Erdős
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  2. Ron Graham
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  3. Imre Z. Ruzsa
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  4. Herbert Taylor
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Additional information

Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901

Research conducted by Herbert Taylor was sponsored in part by the Office of Naval Research under ONR Contract No. N00014-90-J-1341.

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Erdős, P., Graham, R., Ruzsa, I.Z. et al. Bounds for arrays of dots with distinct slopes or lengths. Combinatorica 12, 39–44 (1992). https://doi.org/10.1007/BF01191203

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  • DOI: https://doi.org/10.1007/BF01191203

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