The following may be well-known $-$ but not known to me:

What is the smallest possible size of a set in ${\mathbb F}_2^n$ that blocks every $2$-flat?

Here "blocks" means "have a non-empty intersection with", and $2$-flats are simply affine subspaces of dimension $2$; that is, zero-sum quadruples in ${\mathbb F}_2^n$. Passing to the complements, an equivalent restatement is:

What is the largest possible size of a set in ${\mathbb F}_2^n$ that does not contain any $2$-flat?

For $1$-flats the question becomes trivial: any set with at least two elements contains a $1$-flat.

It is readily seen that any set in ${\mathbb F}_2^n$ containing no $2$-flat has size at most $2^{n-2}$; can this bound be replaced with $c^n$, for some $c<2$? (Lower bounds of this form follow easily by considering random subsets of ${\mathbb F}_2^n$.)

There has been a number of MO posts on blocking sets, like this one; however, they do not seem to address the question above.