This is a stronger version to Szemerédi's theorem.

Let $C : \mathbb{N}\rightarrow 2^{\mathbb{N}}$ be a choice function such that $C(n)$ is a subset of $\{1,...,n\}$ with size at least $\frac{n}{M}$ for some nonzero constant $M$ that only depends on $C$. Can we guarantee an arithmetic progression in $C(n)$ of length at least $\frac{n}{M'}$ where $M'$ only depends on $M$?

If not, what is a counter example? and how big arithmetic progression (order in terms of $n$) we can guarantee in $C(n)$.