OPEN
This is open, and cannot be resolved with a finite computation.
Let $S\subseteq \mathbb{Z}^3$ be a finite set and let $A=\{a_1,a_2,\ldots,\}\subset \mathbb{Z}^3$ be an infinite $S$-walk, so that $a_{i+1}-a_i\in S$ for all $i$. Must $A$ contain three collinear points?
Originally conjectured by Gerver and Ramsey
[GeRa79], who showed that the answer is yes for $\mathbb{Z}^2$, and for $\mathbb{Z}^3$ that the largest number of collinear points can be bounded.
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Additional thanks to: Terence Tao
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #193, https://www.erdosproblems.com/193, accessed 2026-01-16
