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Claim. Theorem 1 of Tsaturian: for every red-blue coloring of the Euclidean plane, either two red points are at distance $1$, or there are blue points

$$x,\ x+d,\ x+2d,\ x+3d,\ x+4d,\qquad \|d\|=1.$$

No measurability or other regularity is assumed of the coloring. So no coloring avoids both a red unit pair and a blue five-term unit progression, and the least avoiding length $K_*$ of Problem 188 satisfies $K_*\ge6$.

Covers. The lower bound $K_*\ge6$: no coloring of the plane avoids both a red unit pair and a blue five-term unit-step progression. The least avoiding length itself stays open.

Proof. The published proof forces colors through small configurations of a unit triangular lattice, classifies the surviving colorings of the lattice into two periodic patterns, and refutes them with an off-lattice choice of a unit chord on a circle of radius $5$. The library's Theorem 1 page writes the argument out with its lemmas.

Postings. The arXiv preprint 1703.10723 was posted on 31 March 2017 (v2 on 4 April 2017), and the paper appeared in the Electronic Journal of Combinatorics 24(4) (2017), #P4.35, published on 24 November 2017. The published version corrects a triangle side length in a lemma of the arXiv version.

Acceptance. Refereed: the Electronic Journal of Combinatorics, 24(4) (2017), #P4.35. The site's commentary credits Tsaturian with the bound, but the site labels the problem OPEN, so that credit is not listed as reviewed. No formalization of the theorem is recorded.

Depends on. Tsaturian, Theorem 1.