Improved Bound for the Gerver-Ramsey Collinearity Problem

Abstract

Let $S$ be a finite subset of $\mathbb{Z}^n$. A vector sequence $(\mathbf{z}_i)$ is an $S$-walk if and only if $\mathbf{z}_{i+1} - \mathbf{z}_i$ is an element of $S$ for all $i$. Gerver and Ramsey showed in 1979 that for $S\subset \mathbb{Z}^3$ there exists an infinite $S$-walk in which no $5^{11} + 1=48{\small,}828{\small,}126$ points are collinear. Here, we use the same general approach, but with the aid of a computer search, to improve the bound to $189$.