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Marshall Hall, Jr., Cyclic projective planes, Duke Math. J. 14 (1947), no. 4, 1079–1090. In the paragraph following his Theorem 4.3, Hall shows that the five residues $-8,-6,0,1,4$, a Sidon set, are contained in no perfect difference set, whatever the modulus. Translating by $9$ gives the Sidon set $\{1,3,9,10,13\}\subset\mathbb{N}$ with the same property, since a translate of a perfect difference set is one. Problem 707 asks whether every finite Sidon set of natural numbers extends to a perfect difference set modulo $p^2+p+1$ for some prime $p$; Hall's obstruction holds for every modulus, so the answer is no and the claim is a full disproof. It also refutes the weaker form Erdős asked elsewhere, with a modulus $m^2+m+1$ for any $m$.

Hall did not state the remark as a theorem or as an answer to a question: Erdős first posed the problem about thirty years later. The identification is due to Alexeev and Mixon (card), who found the passage while preparing their own counterexample, reproved it through Hall's projective-plane and polarity argument (their Theorem 26), and had the statement verified in Lean (not_erdos_707H for the integer set and not_erdos_707H2 for its translate, in the ancillary Lean file posted with version 1 of arXiv:2510.19804 on 2025-10-22 and kept, reformatted, in version 2, the first formalization link; Alexeev re-posted the version-1 file in his repository of formalized Erdős problems on 2025-11-27, the second formalization link); the mechanics of that file are described on the Alexeev–Mixon claim page. They note that Guy's section C10 cites Hall's paper two sentences before stating the question.

Acceptance. The paper is refereed (Duke Math. J.). The site's curator, T. F. Bloom, records Hall's set as the first disproof on the problem page (last edited 2025-10-26), which is the reviewed evidence named here. The month of the issue, December 1947, supplies the page's date. This corpus has not built the Lean file.

Depends on. Nothing beyond the cited paper.