Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 656

claims/: The 1 claim page of Problem 656, one per claimant's result; the problem's standing derives from them.

Statement. Let $A\subset \mathbb{N}$ be a set with positive upper density. Must there exist an infinite set $B\subseteq A$ and integer $t$ such that

\{b_1+b_2: b_1\neq b_2\in B\}+t\subseteq A?

Status. Proved. The status-defining source is Theorem 1.2 of Kra, Moreira, Richter and Robertson [KMRR24] (Commun. Amer. Math. Soc. 4 (2024), 480--494, refereed), which proves the statement for every set of positive upper Banach density, of which positive upper density along the intervals is the special case; the paper calls it Erdős's $B+B+t$ conjecture. The claim page is Kra, Moreira, Richter and Robertson (accepted on the refereed publication and the site's credit).

Source. erdosproblems.com/656, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #656, https://www.erdosproblems.com/656.

References.

[Er75b] Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974), Astérisque 24--25 (1975), 295--310.
[KMRR24] Kra, Bryna and Moreira, Joel and Richter, Florian K. and Robertson, Donald, A proof of Erdős's $B+B+t$ conjecture. Commun. Amer. Math. Soc. 4 (2024), 480--494, doi:10.1090/cams/34.

Formalization. No statement file for the problem is in formal-conjectures, and the community database (teorth/erdosproblems) records formalized "no" (both). The development src/latest/ErdosProblems/Erdos656.lean of Boris Alexeev's lean-proofs repository (first added 2026-08-18; formal authors Codex and GPT-5.6 Sol) declares itself a formalization of Kra, Moreira, Richter and Robertson's solution and is linked on their claim page; this corpus has not built it.

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Linked library material

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Linked from (13)

Kra, Moreira, Richter and Robertson prove the B + B + t conjecture problems/additive_combinatorics/E0656/claims Sumsets and Arithmetic Progressions Sumsets and Arithmetic Progressions additive_combinatorics/erdos_1975_problems_results_combinatorial_number_theory additive_combinatorics/kra_2024_proof_erdos_s_conjecture Corollary 1.3 (p. 2): a set of even integers of positive upper Banach density contains {b_1 + b_2 : b_1 ≠ b_2 ∈ B} for an infinite B, with no shift Theorem 1.2 (p. 2): a set of positive upper Banach density contains a shift of {b_1 + b_2 : b_1 ≠ b_2 ∈ B} for an infinite B Theorem 1.4 (p. 2): Erdős progressions starting at a generic point, the dynamical form of the B + B + t theorem additive_combinatorics/moreira_2019_proof_sumset_conjecture_erdos Question 6.2: whether every set of positive upper density contains t + (B ⊕ B) for an infinite B primes/tao_2023_infinite_partial_sumsets_primes Theorem 1.3: under Dickson-Hardy-Littlewood, an infinite set B of primes with b + b' + 1 prime for distinct b, b'

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