Folkman's conjecture on subset sums and infinite arithmetic progressions

Let $A$ be a strictly increasing sequence of positive integers, and write $A(n):=|A\cap[n]|$, where $[n]:=\{1,2,\dots,n\}$. Thus $A$ has asymptotic density at least $C\sqrt n$ when $A(n)\ge C\sqrt n$ for all sufficiently large $n$. Folkman's conjecture. For any sufficiently large constant $C$, if $A(n)\ge C\sqrt n$ for all sufficiently large $n$, then the subset-sum set $S_A$ contains an infinite arithmetic progression. Folkman proved this under the stronger assumption $A(n)\ge n^{1/2+\epsilon}$ for every arbitrarily small positive constant $\epsilon$, and later work of Szemerédi and Vu proved the full conjecture. Therefore the conjecture is solved.