Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 194

claims/: The 1 claim page of Problem 194, one per claimant's result; the problem's standing derives from them.

Statement. 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?

Status. 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 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.

Source. erdosproblems.com/194, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #194, https://www.erdosproblems.com/194.

References.

[ABJ11] Ardal, Hayri and Brown, Tom and Jungić, Veselin, Chaotic orderings of the rationals and reals. Amer. Math. Monthly (2011), 921-925.

Formalization. Statement in formal-conjectures, which at its commit of 2026-10-06 is marked solved with a formal_proof link to the Lean file listed on the claim page.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (8)

problems/additive_combinatorics/E0194/claims Sumsets and Arithmetic Progressions additive_combinatorics/ardal_2011_chaotic_orderings_rationals_reals Remark 1 (p. 4): for each k at least 2, an ordering of the reals with monotonic k-term but no (k+1)-term progressions Theorem 2.2 (p. 2): an explicit chaotic linear ordering of the integers Theorem 3.1 (p. 3): the rationals have a chaotic linear ordering Theorem 4.1 (p. 4): the reals have a linear ordering with no monotonic 3-term progression additive_combinatorics/erdos_1979_old_new_problems_results_combinatorial_number

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