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Claim. For every $k\ge 3$ there is a linear ordering of $\mathbb{R}$ with no monotone $k$-term arithmetic progression, so the answer is no for every $k\ge 3$.

[ABJ11] calls a linear ordering $\prec$ of a set $X\subseteq\mathbb{R}$ chaotic when no distinct $x,y,z\in X$ with $y=\tfrac12(x+z)$ satisfy $x\prec y\prec z$, and proves (Theorem 4.1) that $\mathbb{R}$ has a chaotic linear ordering. The ordering is built in three steps: the doubling recursion $A_1=\langle 0,-1\rangle$, $A_{n+1}=(2A_n)(2A_n+1)$ gives a chaotic ordering of $\mathbb{Z}$ (Theorem 2.2); König's lemma transfers it to $\mathbb{Q}$ (Theorem 3.1); and with a basis of $\mathbb{R}$ over $\mathbb{Q}$, two reals are compared by the rational ordering at the first basis coordinate where they differ, and a progression $a+c=2b$ holds coordinatewise, so a monotone progression in $\mathbb{R}$ would give one in $\mathbb{Q}$. A monotone $k$-term progression contains a monotone three-term one, so the ordering has none of any length $k\ge 3$. The proof uses the axiom of choice through the existence of a basis; the paper asks whether a choice-free construction exists. Remark 1 of the paper adds, without written proof, that for each $k\ge 2$ some ordering of $\mathbb{R}$ has monotone $k$-term but no $(k+1)$-term progressions. The library card records the paper.

Acceptance. Refereed: Ardal, H., Brown, T. and Jungić, V., Chaotic orderings of the rationals and reals, Amer. Math. Monthly 118 (2011), no. 10, 921–925 (the December 2011 issue, the date of this page). Reviewed: the site's curator, Thomas Bloom, labels the problem disproved, states the negative answer for every $k\ge 3$ and credits it to [ABJ11]. The site's label DISPROVED (LEAN) and the catalog statement Erdos194.erdos_194, marked solved with a formal_proof link, point at a Lean file written with Aristotle and posted by a forum user in the site's discussion thread on 2026-04-15, a formalization of this result linked above; it follows the paper's construction and states its own erdos_194, the existence of a linear ordering of $\mathbb{R}$ with no strictly increasing or decreasing $k$-term progression for every $k\ge 3$, in its own vocabulary rather than the catalog's. The file was not built or audited by this corpus, so formalized is not listed.

Depends on. No wiki page; the claim rests on the cited paper.