Bounds for arrays of dots with distinct slopes or lengths

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 satisfiesnCn 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 1/2Dn 4/5

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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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