Erdős Problem #194 — Must any ordering of the reals contain a monotone $k$-term arithmetic progression for every $k$?

The question, asked by Erdős and Graham, asks whether every linear ordering of the real numbers contains a monotone $k$-term arithmetic progression. Ardal, Brown, and Jungić answered it negatively already for $k=3$.

Known results

• Ardal, Brown, and Jungić, 2011: constructed chaotic linear orderings of $cmathbb{Z}$, $cmathbb{Q}$, and $cmathbb{R}$ with no monotone $3$-term arithmetic progression; the real-number construction uses the axiom of choice.

2011 counterexample

The construction settles the stated problem negatively: not every linear ordering of $cmathbb{R}$ contains a monotone $3$-term arithmetic progression. A later formal-conjectures entry records the result as false, but its displayed Lean theorem still contains by sorry.

Current status (as of March 2026): The problem is resolved negatively, with a published construction for $k=3$; no further issue remains for the stated conjecture.