Szemerédi: dense integer sets contain long arithmetic progressions

What changed

Proved positive-upper-density sets contain arbitrarily long finite arithmetic progressions.

Scientific scope

The record is Szemerédi’s sole-author 1975 Acta Arithmetica paper, pp. 199–245. Roth’s three-term case, Szemerédi’s earlier four-term result, later ergodic and hypergraph proofs, and Green–Tao’s theorem for the primes remain separate.

Evidence

The linked sources identify this contribution and its scope. WRS has not independently reconstructed the scientific claims. The record is Szemerédi’s sole-author 1975 Acta Arithmetica paper, pp. 199–245. Roth’s three-term case, Szemerédi’s earlier four-term result, later ergodic and hypergraph proofs, and Green–Tao’s theorem for the primes remain separate.

Contributors

• Endre Szemerédi · SCIENCE: Named source author of the scoped work; individual effort shares are not measured.

• IMPAN record for Szemerédi theorem paper
• Rényi Institute Szemerédi publication list

Provisional WRS assessment, edition 2026-10-16.1. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.

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