= Dense three-term progressions force four-term progressions
For every $c>0$, all sufficiently large finite integer sets $A$ with at least $c|A|^2$ ordered three-term <arithmetic progressions> contain a nonconstant four-term <arithmetic progression>. <Cauchy-Schwarz inequality> gives $E(A)\ge c^2|A|^3$. 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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