Claims

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1980_03_01_erdos: Erdős's 1980 sequences with at most C representations per sum that no finite partition splits below C, for C = 2, 3, every power of two and every half central binomial coefficient, by Ramsey's theorem; refereed.

1985_01_01_nesetril_rodl: For every integer C at least 2 there is a set of naturals whose pairwise sums each have at most C representations and which no partition into finitely many parts reduces to fewer than C representations in every part; answers no.

2026_06_18_axiommath: Answers the site's wording (ordered pairs, under which C = 2 fails trivially), not the corrected Statement (sums of two distinct elements), so it does not count toward the problem's standing. A Lean 4 development by AxiomProver, published by AxiomMath, refutes that wording by the powers of two at C = 2.

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