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Claim. Jacob Fox and Zach Hunter, in a preprint posted to arXiv on 2026-06-01 ([[../library/additive_combinatorics/fox_2026_three_color_van_der_waerden_numbers/_index|source card]]), prove their Theorem 3,

$$H(k)\ \ge\ k^{(1-o(1))\,k\log k},$$

which gives $H(k)^{1/k}/k\ge k^{(1-o(1))\log k}\to\infty$ and so answers the displayed question of Problem 190, as its precise Statement reads it, with a rainbow $k$-term progression, in the affirmative. The bound follows from the pigeonhole reduction $H(k)\ge w(k;k-1)$ and their Theorem 2, the many-color van der Waerden bound $w(k;r)\ge r^{(1-\varepsilon)k\log k}$ for $k\ge k_0(\varepsilon)$ and $r\ge(\log k)^{3/\varepsilon}$, which rests on a probabilistic construction of very dense subsets of cyclic groups far from containing a $k$-term progression and a random shifted product of colorings. The paper also observes (Section 1.1 and Section 6) that the weaker statement $H(k)=k^{\omega(k)}$ already follows from a product coloring together with either its own three-color bound or the construction of Hunter 2025; the site's commentary records this remark as well. The paper credits Bae's earlier and weaker bound, which has its own claim page, as an independent resolution.

Depends on. No page of this wiki: the proof is self-contained in the preprint apart from the published results it cites.

Acceptance. Reviewed: the site's curator (T. F. Bloom) labels Problem 190 solved and, in the commentary last edited 2026-06-02, states the bound $H(k)\ge k^{(1-o(1))k\log k}$ as Fox and Hunter's. Bae, in a discussion-thread post of 2026-09-14, acknowledges that Fox and Hunter obtained the stronger bound and contests the commentary's attribution of the $H(k)=k^{\omega(k)}$ observation to Hunter's 2025 paper, asking that it be credited to this preprint. The preprint has no refereed publication and no formalization of the result is recorded, so the evidence is reviewed only. The preprint is also the source of the three-color super-exponential bound recorded on Problem 138, which is a different result and not part of this claim.