Vol. 83, No. 2, 1979

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On certain sequences of lattice points

Joseph Leonide Gerver and Lawrence Thom Ramsey

Vol. 83 (1979), No. 2, 357–363

Let S be a finite subset of Rn. A sequence {zi} is an S-walk if and only if zi+1 zi is an element of S for all i. In an effective manner it is shown that long S-walks in Z2 must have an increasing number of collinear points. In Z3, however, an infinite S-walk may have a bounded number of collinear points.

Mathematical Subject Classification 2000
Primary: 10E99, 10E99
Secondary: 05B99
Received: 28 February 1978
Published: 1 August 1979
Joseph Leonide Gerver
Lawrence Thom Ramsey