Claims

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1986_09_08_alon: Alon's Corollary 2.6 (J. Number Theory, 1987): the largest subset of {1,...,n} with no subset summing to 2n has (1/3+o(1))n elements, so the extremal size at c = 1/2 is (1/6+o(1))n along even n; refereed.

2026_07_23_principia_math: Principia Math's 2026 manuscript and Lean: for 0 < c < 1 the normalized extremal size does not converge, and for c at least 1 it equals floor(cn) minus ceil(n/2); filed as full: the second question, and the first for c >= 1.

2026_07_25_beyer_de_ryke: A dated note of 25 July 2026 posted under Principia Math's claim: for 0 < c < 1 the normalized extremal size does not converge, with explicit limits along arithmetic subsequences, and the exact formula for c at least 1.

2026_08_08_principia_math: Principia Math's second result: a dense subset of [1, xT] has a subset summing to sT, so for c = x/s the extremal size is (c/snd(s) + o(1)) n along multiples n of s not divisible by snd(s); Lean states the s = 2 bound for 1/3 < c < 1/2.

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