Claims

../

2008_11_09_nguyen_vu: Theorem 1.4 of Nguyen and Vu (Squares in sumsets, 2010): a square-sum-free subset of the first N integers has at most N^{1/3} (log N)^C elements, so the largest has size N^{1/3+o(1)}; the site's curator credits it as the solution.

2026_08_27_alexeev: A Lean development in Alexeev's repository, with Codex and GPT-5.6 Sol as formal authors, proves SF(N) at most a constant times N^{1/3}(log log N)^16 without Nguyen and Vu's Lemma 4.2, giving the order N^{1/3+o(1)}.

---