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Claim. The first question of Problem 1187 is answered yes for every $k\ge3$: in any coloring of the integers with finitely many colors there is a monochromatic $k$-term arithmetic progression of primes. The claimed result is Theorem 1.2 of Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions: every set of primes of positive relative upper density contains infinitely many $k$-term arithmetic progressions for every $k$. With $r$ colors, some color class contains at least a $1/r$ fraction of the primes up to $N$ for infinitely many $N$, so it has positive relative upper density and the theorem gives the progression. The proof combines Szemerédi's theorem with a transference principle, which carries a density result from a pseudorandom measure to a dense subset of its support, and the Goldston–Yıldırım sieve estimates, which place the primes inside such a measure concentrated on almost primes. The source card is green_2008_primes_contain_arbitrarily_long_arithmetic_progressions.

Covers. The first question alone: monochromatic progressions of primes. The second question, whether a finite coloring of the integers must contain a monochromatic $k$-term progression whose common difference is a prime, is not part of Green and Tao's theorem; it is answered no by the site's own modulo-$4$ coloring, a pending partial claim on its claim page, and by Kenta Kitamura's Lean 4 proof of that coloring on its own page.

Depends on. Nothing in this wiki.

Acceptance. Refereed publication: Ann. of Math. (2) 167 (2008), no. 2, 481--547, doi:10.4007/annals.2008.167.481, issued 1 March 2008; the acceptance rests on this publication alone. The site's curator, Thomas Bloom, labels the problem solved and credits [GrTa08] for the first question in the problem page's commentary (page last edited 8 April 2026), but that credit is not counted as review here, because the curator is also a claimant on this problem, with the modulo-$4$ coloring that answers the second question on its claim page. The arXiv version was posted 8 April 2004, the date of this page. The lean-proofs file linked above, in Boris Alexeev's repository and added on 17 August 2026, declares itself a Lean formalization of a solution to the problem with Green and Tao as its informal authors and Codex and GPT-5.6 Sol as its formal authors; its theorem erdos_1187 proves the first answer from the repository's own Green–Tao theorem together with van der Waerden's theorem obtained through Hales–Jewett, and the second answer by the modulo-$4$ coloring. This corpus has not built or audited it, so it gives no formalized evidence. Kitamura's development is not a formalization of Green and Tao's theorem and is not acceptance evidence for this claim. The accepted Green–Tao claim on Problem 219's claim page records the same theorem for the primes themselves.