For every positive density and positive integer , every sufficiently large subset of of size at least contains a nonconstant arithmetic progression of length .
For every , all sufficiently large finite integer sets with at least ordered three-term arithmetic progressions contain a nonconstant four-term arithmetic progression. Cauchy-Schwarz inequality gives . The Balog-Szemerédi-Gowers theorem supplies a large small-doubling subset. A bounded-modulus Freiman 2-isomorphism gives a positive-density cyclic model; apply the assumed four-term case of Szemerédi's theorem to representatives, then lift the two consecutive second-difference relations.
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Szemerédi's theorem is a fundamental result in combinatorial number theory which pertains to arithmetic progressions in sets of integers. Specifically, the theorem states that for any positive integer \( k \), any subset of the integers with positive density contains a non-trivial arithmetic progression of length \( k \). More formally, if \( A \) is a subset of the positive integers with positive upper density, i.e.