Improved bound for the Gerver-Ramsey collinearity problem

Abstract

Let S be a finite subset of Zn. A vector sequence (zi) is an S-walk if and only if zi+1−zi is an element of S for all i. Gerver and Ramsey showed in 1979 that for S⊂Z3 there exists an infinite S-walk in which no 511+1=48,828,126 points are collinear. Here, we use the same general approach, but with the aid of a computer search, to improve the bound to 189.