Sidon Complements and Progression-Hitting Constructions

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baire_construction: Uses Baire category to obtain uncountably many Sidon sets meeting every infinite progression.

bloom_2026_sidon_complement_constructions: Identifies the dated public constructions, their attribution limits, and the linked formal source.

diagonal_construction: Builds a lacunary set by selecting one sufficiently large point from each infinite progression.

factorial_construction: Verifies the factorial-plus-index Sidon set attributed to AlphaProof and its progression-hitting property.

lacunary_sidon: Proves uniqueness of two-term sums in a positive sequence whose successive terms at least double.

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The source record identifies the inspected public problem page, Dutta's comment, and the linked AI-assisted formal source. This is a web source unit; no canonical PDF exists for those online expositions. The capture discussion_20260905.html.gz (the page titled "198 Discussion Thread | Erdős Problems", snapshotted 2026-09-05) contains no copyright, license or terms line, the captured pages' only reuse-related text being a recommended citation format, and the site's homepage and FAQ (https://www.erdosproblems.com/ and https://www.erdosproblems.com/faq, read 2026-10-02) state no copyright, license or terms of use; the term is unstated. The capture problem_20260905.html.gz (the page titled "198 | Erdős Problems", snapshotted 2026-09-05) contains no copyright, license or terms line, and the same site pages state no terms; the term is unstated. The capture proof_claims_20260905.html.gz (the page titled "Erdős Problems", snapshotted 2026-09-05) contains no copyright, license or terms line, and the same site pages state no terms; the term is unstated.

All three constructions disprove Problem 198. A common doubling-gap lemma supplies the Sidon property. The diagonal construction chooses one large point from each enumerated progression. The AlphaProof-attributed construction encodes residues explicitly. The Baire-category argument gives uncountably many counterexamples through modular conditions on power sequences. The shared lemma is proved once; the different hitting arguments are preserved separately.

Compilation scope. The four complete natural-language proof components are author-recorded. An independent review on 2026-09-05 is reported, but its report is not filed with this source and supplies no independent-review credit here. The historical Baumgartner paper was obtained and inspected on 2026-09-06. It contains the related successive-selection method, but no explicit Sidon theorem; the historical source record preserves the remaining attribution qualification. The reported formal build has not been reproduced.

Bears on. Problem 198: the diagonal, factorial and power-sequence constructions each give a Sidon set of positive integers meeting every infinite arithmetic progression, so its complement contains none, a negative answer to the question; the doubling-gap lemma is the Sidon step common to all three. The real-set problem is contextual: it asks whether every subset of $\mathbb R$ with no 3-term arithmetic progression has an infinite arithmetic progression in its complement. Baumgartner's 1975 theorem answers no, with a subset of $\mathbb R$ that has no 3-term progression and meets every infinite one; none of the integer Sidon constructions here gives such a set.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.