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Claim. For a function $f$ from the positive integers to the positive reals, let $w(f,3,2)$ be the least $w$ such that every $2$-coloring of $\{1,\ldots,w\}$ has a monochromatic three-term arithmetic progression $a,a+d,a+2d$ with $d\ge f(a)$. Brown and Landman's Theorem 7 states that $w(f,3,2)$ exists for every such $f$. With $f(a)=a+1$ the condition $d\ge f(a)$ is $d>a$, so every $2$-coloring of the positive integers has a monochromatic $x,x+d,x+2d$ with $d>x$, which answers Problem 645 yes; the finite form and the form on all of $\mathbb N$ are equivalent by compactness, and the paper's first proof establishes the infinite form directly. The theorem is paged at Theorem 7 of the library's source card. The paper's stronger version of the theorem gives an explicit bound on $w(f,3,2)$ for non-decreasing $f$, and its Theorem 12 shows that the statement fails for four-term progressions and for more than two colors, so the three-term, two-color case is the whole content of the question.

Argument. The first proof (half a page) reduces to non-decreasing $f$, reads a $2$-coloring as a binary sequence, and either finds a constant or alternating tail, in which a sufficiently late progression of suitable parity serves, or finds two occurrences of the pattern $001$ (or, symmetrically, $110$) at a distance $d\ge f(x+2)$ and reads off one of the progressions $\{x+2,x+d+2,x+2d+2\}$ or $\{x,x+d+1,x+2d+2\}$ as monochromatic with a large enough difference; compactness then gives the finite $w$.

Depends on. Nothing in this wiki; the result is the paper's own theorem.

Dating. The page is dated by the issue month of the journal record (Bull. Austral. Math. Soc. 60 (1999), no. 1, August 1999, per the Crossref record, whose only online date, 17 April 2009, is the digitization); the day in the page name is a placeholder, and the paper link carries no date for that reason.

Acceptance. Reviewed: the site's curator, T. F. Bloom, credits the first proof of the problem to Brown and Landman's paper in the problem's commentary and labels the problem PROVED (LEAN) (page last edited 4 April 2026); the curator is independent of the authors. Refereed: Bull. Austral. Math. Soc. 60 (1999), no. 1, 21--35. The community database also lists the problem as proved. The site's own elementary argument, attributed to Ryan Alweiss, is a second proof with its own claim page.

Read depth. The paper is held as the authors' 14-page copy with its own pagination, not the journal text; Theorem 7 (p. 6 of the copy) was read clause by clause on the page image and its first proof was followed, not independently reviewed; the stronger version (p. 7) and Theorem 12 (pp. 10--11) were read at statement depth. The specialization $f(a)=a+1$ is an authored step of one line. Nothing is independently reviewed in this corpus.