= Solution
Part (iii) gives at least $\eta^2|A|^3$ <additive quadruple>[additive quadruples]. By the <Balog-Szemerédi-Gowers theorem>, there is $A'\subseteq A$ with
$$
|A'|\ge c(\eta)|A|,
\qquad
|A'+A'|\le K(\eta)|A'|.
$$
The <Freiman-Ruzsa theorem> places $A'$ in a coset progression $P$ whose rank is bounded in terms of $\eta$ and which satisfies
$$
|P|\le C_1(\eta)|A'|.
$$
Thus $A'$ has density at least $C_1(\eta)^{-1}$ inside $P$.
The <Szemerédi theorem in a bounded-rank coset progression> says that, for fixed rank and density, every sufficiently large such progression has a nontrivial four-term progression in every subset of that density. If $|A|\ge C(\eta)$ for a sufficiently large $C(\eta)$, then $|P|\ge|A'|\ge c(\eta)|A|$ crosses this threshold. It gives a nontrivial four-term progression in $A'$, which is also contained in $A$. This proves the claim with a constant depending only on $\eta$.
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