Problem 194

Let $k\geq 3$ . Must any ordering of $\mathbb{R}$ contain a monotone $k$ -term arithmetic progression, that is, some $x_1<\cdots<x_k$ which forms an increasing or decreasing $k$ -term arithmetic progression?

Source: erdosproblems.com/194

Arithmetic progressions

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An accepted solution exists. The statement is false.

DISPROVED (LEAN), the site's label: the answer is no for every $k\ge 3$ , by the ordering of $\mathbb{R}$ with no monotone three-term arithmetic progression of Ardal, Brown and Jungić [ABJ11], recorded on its claim page (Ardal, Brown and Jungić, 2011) with its refereed and site evidence; the label's Lean mark refers to a Lean file posted in the site's discussion thread in April 2026 and linked by the catalog, listed on the claim page and not built here.

Problem 194

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