  Using various results from extremal set theory (interpreted in the language
of additive combinatorics), we prove an asyptotically sharp version of
Freiman's theorem in F_2^n: if A in F_2^n is a set for which |A + A| <= K|A|
then A is contained in a subspace of size 2^{2K + O(\sqrt{K}\log K)}|A|; except
for the O(\sqrt{K} \log K) error, this is best possible. If in addition we
assume that A is a downset, then we can also cover A by O(K^{46}) translates of
a coordinate subspace of size at most |A|, thereby verifying the so-called
polynomial Freiman-Ruzsa conjecture in this case. A common theme in the
arguments is the use of compression techniques. These have long been familiar
in extremal set theory, but have been used only rarely in the additive
combinatorics literature.
