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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 774.05020

**Autor: ** Erdös, Paul; Graham, Ron; Ruzsa, Imre Z.; Taylor, Herbert

**Title: ** Bounds for arrays of dots with distinct slopes or lengths. (In English)

**Source: ** Combinatorica 12, No.1, 39-44 (1992).

**Review: ** Authors' abstract: An n× m sonar sequence is a subset of the n× m grid with exactly one point in each column, such that the {m\choose 2} vectors determined by them are all distinct. We show that for fixed n the maximal m for which a sonar sequence exists satisfies n-Cn^{11/20} < m < n+4n^{2/3} for all the n and m > n+c log n log log n for infinitely many n.

Another problem concerns the maximal number D of points that can be selected from the n× m grid so that all the \binom{D}{2} vectors have slopes. We prove n^{ ½} << D << n^{4/5}.

**Reviewer: ** J.R.Seberry (Lincoln)

**Classif.: ** * 05B15 Orthogonal arrays, etc.

05B30 Other designs, configurations

**Keywords: ** arrays of dots; sonar sequence; slopes

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag