Erdős Problem #193

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

The allowed step set is one fixed finite subset of ${\mathbb{Z}}^3$. The set of visited vertices must be infinite, so repeated circulation around a finite loop is not an example. Collinearity means lying on one affine real line, not necessarily a coordinate line. [S1] [S2]

Statement recorded in the supplied source.

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? [S1]

Short English statement

Must an infinite three-dimensional lattice walk using only finitely many permitted step vectors visit three collinear points?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1985 (EP-193), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 193, https://www.erdosproblems.com/193. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Joseph L. Gerver and L. Thomas Ramsey, On certain sequences of lattice points, Pacific Journal of Mathematics 83 (1979), 357-363. https://msp.org/pjm/1979/83-2/pjm-v83-n2-p08-p.pdf. Bibliographic information from the supplied record.

• S4 Thomas F. Lidbetter, Improved bound for the Gerver-Ramsey collinearity problem, Discrete Mathematics 347 (2024), Article 113718. https://doi.org/10.1016/j.disc.2023.113718. Bibliographic information from the supplied record.

• S5 Gerver, Joseph L. and Ramsey, L. Thomas, On certain sequences of lattice points. Pacific J. Math. (1979), 357-363. Bibliographic information from the supplied record.