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Claim. In his 1965 survey (display (69), printed pp. 228--229) Erdős states that his conjecture on lcm triples in a sequence of positive lower density would follow from the bound $f(n)=o(2^n)$ for the maximum size of a union-free family of subsets of an $n$-set, reports that Sárközy and Szemerédi had proved this bound, and indeed $f(n)<c2^n/\log\log n$, in unpublished work, and concludes that the conjecture about triples is thereby proved. The claim is Erdős's, since he published it: the answer yes to this problem, resting on his reduction, which the survey asserts and does not carry out, and on a bound whose authors published no account of it.

Postings. The survey, Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III, Wiley (1965), 196--244, linked above through its scan in the Rényi Institute's Erdős archive; the page is dated to the first day of 1965, the survey's year. Sárközy and Szemerédi published nothing on the bound, so the only record of it is Erdős's report. The site's page for Problem 447 records the report in the same terms; the site credits the problem to Kleitman's published theorem instead.

Standing. Claimed, not accepted: the bound was never published and no record of its review exists, and the survey states the reduction without proof. The bound itself is true, since Kleitman's published theorem (the sibling page Kleitman) is sharper, but the standing of a claim is that of its own evidence. Nothing was reviewed here.

Depends on. No page of this wiki.