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Claim. There is a Sidon set $A\subset\mathbb{N}$ meeting every infinite arithmetic progression, so its complement contains no infinite arithmetic progression and the answer is no.

Enumerate the countably many infinite arithmetic progressions in $\mathbb{N}$ as $P_1,P_2,\ldots$. Take $a_1$ the least element of $P_1$ and, for $n\ge 2$, $a_n$ an element of $P_n$ with $a_n>2a_{n-1}$. The set $A=\{a_1<a_2<\cdots\}$ meets every $P_n$, and since each term exceeds twice the one before, two-term sums are distinct, so $A$ is Sidon. The library's construction page writes the argument out with its doubling-gap lemma.

Attribution. Erdős and Graham's survey [ErGr79] (printed p. 339, and their 1980 book in the same words) credits Baumgartner with the positive assertion, that the complement of a Sidon set contains an infinite arithmetic progression. The site's page gave that positive answer, citing [Ba75], until some time between November 2024 and mid-May 2025: web archive captures of 2024-06-19 and 2024-11-07 show it labeled solved with the answer yes, as shown by Baumgartner, and the formal-conjectures catalog's statement file of 2025-05-01 stated the answer yes on the remark of Erdős and Graham (1980, p. 23), noting in a lemma that the curator cites [Ba75] for that assertion and that it runs opposite to Baumgartner's theorem. After Google DeepMind reported AlphaProof's counterexample to the curator, which the catalog's issue of 2025-05-13 and pull request of 2025-05-15 record, the site switched to the answer no, kept the credit to Baumgartner on the report of Erdős and Graham, said that [Ba75], Baumgartner, J. E., Partitioning vector spaces, J. Combin. Theory Ser. A 18 (1975), no. 2, 231–233 (the March 1975 issue, the date of this page), does not state it exactly, and wrote out the construction above as implicit in it. The paper proves that a vector space over $\mathbb{Q}$ has a subset meeting every infinite arithmetic progression and containing no three-term one, the statement behind Problem 199, by selecting a point in each enumerated progression beyond all earlier coefficient magnitudes, the same successive selection; it contains no Sidon statement, and the source record preserves the survey's contradictory sentence. The page is named for Baumgartner because the site credits him; Baumgartner never stated the integer result, and as a disproof of this problem the construction was first written out on the site's page, by mid-May 2025.

Acceptance. None. The site's curator, Thomas Bloom, labels the problem disproved and credits the result to Baumgartner on the report of Erdős and Graham, with the qualification that [Ba75] does not state it exactly. But the curator wrote out the integer construction above himself (the formal-conjectures catalog's pull request of 2025-05-15 credits the elementary argument to him), Baumgartner never stated it, and Erdős and Graham credit Baumgartner with the opposite answer. The label therefore accepts the curator's own write-out and is not acceptance independent of the claim. No refereed publication states the integer result, and the natural-language proof on the library page is author-recorded. The problem's standing rests on the accepted claims of Google DeepMind and Dutta.

Depends on. No wiki page; the claim rests on the construction stated above.