Claims

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1968_08_01_berlekamp: Berlekamp's Galois-field two-coloring shows W(p+1) exceeds p times 2 to the p for prime p, which with the trivial comparison to log W(k) gives f(k) at least (log 2 over 2 minus o(1)) times k, the bound the site credits to him.

1977_02_01_gerver: Gerver proved that the largest reciprocal sum of a set with no k-term arithmetic progression is at least (1-o(1)) k log k, and that the finiteness of every f(k) is equivalent to Erdős's reciprocal-sum conjecture.

1984_07_01_wroblewski: Wróblewski constructed an explicit set of integers with no three-term arithmetic progression and reciprocal sum above 3.00849, the record for f(3) that the site credits.

2022_03_11_walker: Walker found digit-restricted Kempner sets with no four-term, respectively ten-term, arithmetic progression and reciprocal sums 4.43975 and 14.056, and showed Kempner sets suffice for f(k); an arXiv preprint, claimed.

2026_09_23_openai: Corollary 11.2 of the OpenAI release manuscript of 23 September 2026 claims that every set with no k-term progression has reciprocal sum at most H_k = Σ 2^{-m} r_k(2^m) < ∞, so f(k) is finite for each k ≥ 3; no numerical bound.

2026_09_26_kiichi: Two explicit finite sets, one free of three-term and one of four-term progressions, with reciprocal sums beyond the records of Wróblewski (1984) and Walker (2025); each stated as the claimants' own Lean theorem, claimed.

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