Claims

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1992_01_01_erdos_sarkozy: Theorem 4 of Erdős and Sárközy (Discrete Math. 1992) bounds the threshold K(N) for a three-term progression among subset sums, which in the problem's notation gives g_3(n) ≫ 3^n/n; refereed and credited by the site.

2026_06_23_korsky: A preprint of 2026-06-23 proving g_3(n) ≥ (√3/(2√π) + o(1)) 3^n/√n by the ternary characterization of progression-free subset sums, with exponential lower and upper bounds for every fixed k ≥ 4; announced in the site's thread.

2026_09_05_costa: Simone Costa's preprint of 2026-09-05, developed with ChatGPT (OpenAI, GPT-5.6 Sol), proves that the liminf of g_3(n)/3^n is zero, so the displayed question of Problem 817 has a negative answer; on the site's tab, so claimed.

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