This question has practical meanings in algebraic attack of stream ciphers in cryptography. It can be stated as follows:

Suppose $V$ is a $n$ dimensional vector space over the field $F_2$, where $F_2$ is the simplest finite field contains only $\{0,1\}$. you can take $V$ to be $F_2^n$. $S$ be a subset of $V$. The question is, What is the minimum of the size of $S$ such that $S$ must contain some $k$ dimensional linear sub-manifold $M$ of $V$, where $k$ is a given positive integer below $n$?

Here by a linear sub-manifold $M$ of dimension $k$, we mean that $M$ can be expressed as $M=\{x+W\}$, where $x$ is a fixed vector in $V$ and $W$ is a $k$ dimensional subspace of $V$.

It's obvious that $|S|>2^k$，since any sub-manifold of dimension $k$ contains exactly $2^k$ points but there are subsets of size $2^k$ that are not submanifolds. I think this problem must has been considered by combinatorists, but I didn't find it in any standard textbooks on combinatorics.