Claims

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1999_04_01_granville_roesler: Theorem 2 of the 1999 Monthly paper: m^{1/2} <= h(m) and h(m) is at most about (3/2)(2m)^{2/3}, the lower bound by a pairing argument credited to Erdős and Szemerédi, the upper by the Freiman–Lev sets; refereed.

2026_06_10_schmitt_gehrunger_dekoninck_berczi_kreitner_price_holmes: Theorem A.1 of the July 2026 ProofCouncil preprint by Schmitt and six coauthors: h(n) is at most n^(1/2) exp(C sqrt(log n)), so the exponent of h(n) is one half; in the site's commentary under its label OPEN, not refereed.

2026_09_05_kitamura: A public Lean 4 development of September 2026 states, for the formal-conjectures threshold, that h(n)/sqrt(n) tends to infinity; not built or audited by this corpus, and stated in no paper.

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