Claims

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1980_06_13_erdos_pomerance: Erdős and Pomerance (Indag. Math. 1980) prove h(n) << n^{3/2}/(log n)^{1/2}, the upper bound the site's commentary records; refereed, partial.

1980_06_13_erdos_selfridge: Erdős and Selfridge's lower bound h(n) > (3-o(1))n, by Brun's method, recorded by Erdős and Pomerance (1980) and by Guy without a printed proof; pending.

1995_01_01_ruzsa: Ruzsa (Studia Sci. Math. Hungar. 1995) builds sets of n primes with few multiples in long intervals, which gives h(n)/n -> infinity; refereed, partial.

2026_07_26_korsky: A proof claim that h(n) is at least n exp((log 2 / 2 - o(1)) log n / log log n), by adapting a quadratic-residue compression argument of Green and Ruzsa; later Theorem 1.3 of a joint arXiv paper with Chen; pending.

2026_07_29_chen: A proof claim that h(n) is at least n exp(c log n / log log n) and at most n^1.4, with F(n) of Problem 711 at most n^1.4031; the second, joint version of the arXiv paper with Korsky sharpens the upper bounds to n^{4/3}; pending.

2026_08_13_chen_korsky: Theorem 1.2 of the second version of Chen and Korsky's arXiv paper: h(n) << n^{4/3}/(log n)^{1/3}, from a new estimate for unions of arithmetic progressions; made with ChatGPT-5.6 Sol; pending.

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