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



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



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> What is the largest possible size of a set in ${\mathbb F}\_2^n$ that does not contain any $2$-flat?
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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](https://mathoverflow.net/questions/222375/n-2-blocking-sets-in-agn-2); however, they do not seem to address the question above.



Sets blocking every $2$-flat in $AG(n,2)$
An $N$-element set contains no 2-flat iff all pairwise sums of its elements are distinct, so ${N\choose 2}\leq 2^n$, whence $N<1+2^{(n+1)/2}$.


**[UPDATE]** It seems that I have a construction providing $2^n$ points in $\mathbb F\_2^{2n}$, confirming that $c=\sqrt2$ is optimal.


The idea is as follows. Denote $U=\mathbb F\_2^n$. Assume that we have a map $\phi\colon U\to U$ such that $0\neq a+b=c+d, \; \{a,b\}\neq\{c,d\}\; \Rightarrow \; \phi(a)+\phi(b)\neq \phi(c)+\phi(d)$. Then the pairs $(a,\phi(a))\in U\oplus U$ form a required set of $2^n$ vectors, because their pairwis sums are distinct. It suffices now to find such $\phi$.


Identify $U$ with $\mathbb F\_{2^n}$. THen $\phi(x)=x^3$ works. Indeed, if $0\neq a+b=c+d$ and $a^3+b^3=c^3+d^3$, then $a^2+b^2=(a+b)^2=(c+d)^2=c^2+d^2$ and $a^2+ab+b^2=\frac{a^3+b^3}{a+b}=\frac{c^3+d^3}{c+d}=c^2+cd+d^2$. Thus $ab=cd$, and the pairs $(a,b)$ and $(c,d)$ are pairs of roots of the same quadratic equation.


