  Let $\mathcal{P}$ denote the set of all primes. $P_{1},P_{2},P_{3}$ are three
subsets of $\mathcal{P}$. Let $\underline{\delta}(P_{i})$ $(i=1,2,3)$ denote
the lower density of $P_{i}$ in $\mathcal{P}$, respectively. It is proved that
if $\underline{\delta}(P_{1})>5/8$, $\underline{\delta}(P_{2})\geq5/8$, and
$\underline{\delta}(P_{3})\geq5/8$, then for every sufficiently large odd
integer n, there exist $p_{i} \in P_{i}$ such that $n=p_{1}+p_{2}+p_{3}$. The
condition is the best possible.
