Problem 193

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?

Source: erdosproblems.com/193

Geometry

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SolvedDisproved

An accepted solution exists. The statement is false.

The site labels the problem DISPROVED (LEAN) on its problem page as accessed 2026-09-08 (OPEN in the site's export of 2026-09-04, the day the curator confirmed the update) and credits Cambie and Kalviainen, assisted by AI, with an infinite bounded-step walk in $\mathbb{Z}^3$ containing no three collinear points [CaKa26], Theorem 1 (Cambie and Kalviainen 2026) of arXiv:2609.01766v1 (submitted 2026-09-01). The derived standing agrees with the label: it rests on the accepted claim page (Cambie and Kalviainen, 2026) of that proof, and the Lean suffix refers to the authors' formalization, which this corpus has not built or audited.

Problem 193

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