Erdős problems frontier

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  "text": "Erdős Problem #198 has status 'disproved (lean)'. Statement: The answer is no; Erdős and Graham report this was proved by Baumgartner, presumably referring to the paper [Ba75], which does not state this exactly, but the following simple construction is implicit in [Ba75]. Let $P_1,P_2,\\ldots$ be an enumeration of all countably many infinite arithmetic progressions. We choose $a_1$ to be the minimal element of $P_1\\cap \\mathbb{N}$, and in general choose $a_n$ to be an element of $P_n\\cap \\mathbb{N}$ such that $a_n>2a_{n-1}$. By construction $A=\\{a_1 < a_2 < \\cdots\\}$ contains at least one element from every infinite arithmetic progression, and is a lacunary set, so is certainly Sidon. AlphaProof has found the following explicit construction: $A = \\{ (n+1)!+n : n\\geq 0\\}$. This is a Sidon set, and intersects every arithmetic progression, since for any $a,d\\in \\mathbb{N}$, $(a+d+1)!+(a+d)\\in A$, and $d$ divides $(a+d+1)!+d$. This was formalized in Lean by Alexeev using Aristotle. Topics: additive combinatorics, sidon sets, arithmetic progressions. Erdős prize: no. Statement is machine-verified in Lean (formal-conjectures). OEIS: N/A.",
  "type": "open_question"
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  "text": "Agent-imported candidate claim; scope requires review."
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 "created": "2026-05-30T00:42:06.826507+00:00",
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  "authors": [
   {
    "name": "Erdős Open-Problem spine ingest",
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   }
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  "extraction": {
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  "title": "cap_61973ee16b553d57 · vc_cb7973c232de2315",
  "url": "https://www.erdosproblems.com/198",
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