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

The ternary Goldbach problem with primes in positive density sets