Theorem

Szemeredi's Theorem

Number Theory

Any set of integers with positive upper density contains arbitrarily long arithmetic progressions. Proved by Endre Szemeredi, it generalizes the earlier Van der Waerden and Erdos-Turan results and underlies the later Green-Tao theorem.

Facts

Statement

A subset of the natural numbers with positive upper density contains an arithmetic progression of length k for every positive integer k. 1

Proof Year

1975 1

Classification

Statement Form

Existence Theorem 1

Connections

Has Statement Form

Existence Theorem

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

In Branch

Number Theory

Sources

1. Szemeredi's Theorem (Wikipedia)

Wikimedia Foundation

Statement section, first sentence
Szemerédi's theorem asserts that a subset of the natural numbers with positive upper density contains an arithmetic progression of length k for all positive integers k.
lead paragraph, third sentence
Endre Szemerédi proved the conjecture in 1975.

Switch Atlas

The Inner World

The Civil World

The Natural World

Szemeredi's Theorem

Has Statement Form

In Branch