Three-collinearity-free walks in dimension four

Determine whether there exists an infinite walk in four-dimensional integer space using only standard unit basis vector steps and containing no three collinear points.

Background

The paper proves the existence of an infinite four-dimensional standard-basis walk with no four collinear points. It additionally gives a finite walk of length 5500 with no three collinear points.

The open problem is whether the three-collinearity-free property can hold for an infinite four-dimensional standard-basis walk.

References

We also cannot rule out an infinite walk in $4$ with no $3$ collinear points.

— Brown-Gerver-Ramsey Theorems in Small Dimensions  (2609.20366 - Cambie et al., 17 Sep 2026) in Section 1, paragraph following the discussion of three-dimensional walks

As we mentioned, we do not know if the result of Theorem~\ref{thm:main} could be improved to $5$ or $4$. The third author found a possible construction in $5$ , but we have not been able to prove that it works.

— Brown-Gerver-Ramsey Theorems in Small Dimensions  (2609.20366 - Cambie et al., 17 Sep 2026) in Section Final words