I have a good motivation to ask the question below, but since the post is already a little long, and the problem looks rather natural and appealing (well, to me, at least), I'd rather go straight to the point.

Let $n\ge 3$ be an integer. If $E$ denotes the standard basis of the
vector space ${\mathbb F}_2^n$, then for any subset $A\subset{\mathbb
F}_2^n$ we have $|A+E|\ge|A|$. This trivial estimate is easy to improve
in various ways, but this is not my concern here. What I am interested in
instead is the sumset $A{\stackrel2+} E$ consisting of all those vectors
of ${\mathbb F}_2^n$ with *at least two* representations as $a+e$,
where $a\in A$ and $e\in E$; that is, vectors at Hamming distance $1$
from at least two elements of $A$. How small can this sumset be?

It is not difficult to find a linear subspace $L<{\mathbb F}_2^n$ of co-dimension ${\rm codim}\,L=\lfloor\log_2 n\rfloor+1$ such that every two elements of $L$ are at least distance $3$ from each other. Clearly, $L{\stackrel2+} E$ is empty, showing that if $|A|<2^n/n$, then, in general, no lower bound for $|A{\stackrel2+} E|$ can be obtained. Let's assume, however, that $A$ is large; what can be said in this case? A simple double counting shows that $$ |A{\stackrel2+} E| > \Big( 1-\frac{2^n}{n|A|} \Big) |A|. $$ My question is: assuming that $A$ is large enough (say, $|A|=2^{n-1}$), can this estimate be improved to $|A{\stackrel2+} E|\ge|A|$? Here is a way to put it in a particularly simple, notation-free form:

Suppose that half of the vertices of the $n$-dimensional hypercube are colored, say, red. Is it true that under any such coloring, at least half of the vertices have two (or more) red neighbors?

Notice that, if true, the estimate $|A{\stackrel2+}E|\ge|A|$ is best possible: equality is attained, for instance, if $A$ is the set of all vectors of the same parity (alternatively, all vectors with the first coordinate equal to $0$, or all vectors with the sum of all coordinates, but the first one, equal to $0$).

Another remark is that $|A{\stackrel2+} E|\ge|A|$ holds true if $A\subseteq{\mathbb F}_2^n$ is an *affine subspace* with $|A|>2^n/n$.