# How large can a non-sumset be?

The theory of sumsets $A+B$ where $A$ and $B$ are finite subsets of an additive group $Z$ is extensively studied in additive combinatorics: finding long arithmetic progressions inside them, finding lots of subsets of this form, bounding its size above and below, and so on.

A fairly natural inverse question is the following.

Is there a function $f$ such that if $\lvert A\rvert\gg f(\lvert Z\rvert)$ then $A=B+C$ where $B$ and $C$ are both fairly large sets?

Since results such as Szemeredi's theorem and Ramsey theory suggest that sets can have lots of structure from cardinality conditions alone, and sumsets are very structured, this seems like a plausible hope.

The case for general finite additive groups may be too hard/trivially false, so I am (as usual in these questions) mostly interested in the cases $Z=\mathbb{Z}/N\mathbb{Z}$ and $Z=\mathbb{F}_p^n$.

I suspect that this sort of result is already known, or follows easily from another well known theorem, and would appreciate any reference and/or proof.

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