Problem 192

Let $A=\{a_1,a_2,\ldots\}\subset \mathbb{R}^d$ be an infinite sequence such that $a_{i+1}-a_i$ is a positive unit vector (i.e. is of the form $(0,0,\ldots,1,0,\ldots,0)$ ). For which $d$ must $A$ contain a three-term arithmetic progression?

Source: erdosproblems.com/192

Arithmetic progressions
Combinatorics

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An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

SOLVED (LEAN), the site's label: the walk must contain a three-term progression exactly when $d\le 3$ . The accepted claim is Keränen's four-letter word without abelian squares, which gives the counterexamples for $d\ge 4$ , together with the finite check that every ternary word of length $8$ contains an abelian square, recorded on its claim page (Keränen, 1992) with its curator and survey evidence; the label's Lean mark follows Luccioli's Lean proof of the full classification posted in the site's thread in May 2026, and the catalog links Alexeev's later file of August 2026, both listed on the claim page and not built by this corpus.

Problem 192

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