Claims

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1965_01_01_erdos: Erdős's Theorem 2 of 1965: any n nonzero reals contain n/3 of them with no sum of two, equal or distinct, equal to a third, so (n - 1)/3 when 0 is in the set (Problem 792); a proceedings volume, no refereeing evidence.

1990_01_01_alon_kleitman: Alon and Kleitman's Proposition 1.1 (1990): any n nonzero integers contain a sum-free subset of more than n/3 elements, so f(n) >= n/3 for n >= 2 in Problem 792; a chapter in a tribute volume, no refereeing evidence.

1997_12_01_bourgain: Bourgain (Israel J. Math. 1997): every set of n >= 3 positive integers has a sum-free subset of at least (n + 2)/3 elements, the best refereed lower bound for positive-integer sets; it does not cover sets with 0 or negative elements.

2013_01_19_eberhard_green_manners: The theorem that some set of n positive integers has no sum-free subset larger than n/3 + o(n), which with Erdős's lower bound n/3 settles the main term of f(n) in Problem 792; Annals of Mathematics 2014, credited by the site.

2025_02_12_bedert: Bedert's 2025 preprint theorem that every finite set A of integers has a sum-free subset of at least |A|/3 + c log log |A| elements, the first lower bound for f(n) exceeding n/3 by an unbounded amount; unrefereed.

2026_09_09_bedert: A proof claim on the site's tab by Bedert, worked out with GPT 6 Astra, that f(n) ≥ n/3 + (log n)^{1/3-o(1)}, improving the log log n of his preprint; an AI-generated write-up on a file-sharing service, unreviewed.

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