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Claim. Theorem 1.2 of Currier, Mody, Xie and Zhang: there is a red-blue coloring of the whole Euclidean plane with no two red points at unit distance and no blue unit-step progression of $6330$ points, in any location or direction. So the least avoiding length $K_*$ of Problem 188 satisfies $K_*\le6330$.

Covers. The upper bound $K_*\le6330$ only. The least avoiding length itself stays open.

Proof. The coloring is random over a periodic hexagonal cell construction at scale $99/100$ with selection probability $p=1/19$, red on selected cells, so every outcome has no red unit pair. Lemma 3.1 counts the cell tuples that can carry a unit progression, and Lemma 3.2 bounds the probability that such a tuple is all blue by $e^{-0.01557m}$; the expected number of all-blue tuples is then below $1$ for $m=6330$. The paper settles both numerical steps by a short calculation that it does not display. The library's Theorem 1.2 page reconstructs the proof, makes the boundary conventions explicit and certifies both numerical steps with exact rational bounds.

Postings. arXiv:2606.17194, v1 of 15 June 2026, which already states the bound $6330$, and v2 of 31 August 2026, which improves the paper's general exponential base to $6.79$ and leaves the planar bound unchanged.

Acceptance. None. No journal publication or outside review is recorded, the site's page does not mention the result, and the library's reconstruction is author-recorded, so the claim stays claimed.

Depends on. Currier, Mody, Xie and Zhang, Theorem 1.2.