  Let $\delta > 1/2$. We prove that if $A$ is a subset of the primes such that
the relative density of $A$ in every reduced residue class is at least
$\delta$, then almost all even integers can be written as the sum of two primes
in $A$. The constant $1/2$ in the statement is best possible. Moreover we give
an example to show that for any $\varepsilon > 0$ there exists a subset of the
primes with relative density at least $1 - \varepsilon$ such that $A+A$ misses
a positive proportion of even integers.
