Vol. 83, No. 2, 1979

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Long walks in the plane with few collinear points

Joseph Leonide Gerver

Vol. 83 (1979), No. 2, 349–355

Let S be a set of vectors in Rn. An S-walk is any (finite or infinite) sequence {zi} of vectors in Rn such that zi+1 zi S for all i. We will show that if the elements of S do not all lie on the same line through the origin, then for each integer K 2, there exists an S-walk WK = {zi}i=1N(K) such that no K + 1 elements of WK are collinear and N(K) grows faster than any polynomial function of K.

Mathematical Subject Classification 2000
Primary: 10E99, 10E99
Secondary: 05B99
Received: 16 June 1978
Published: 1 August 1979
Joseph Leonide Gerver