By a variant of a method of Ramaré and Ruzsa we arrive at a bound
for the additive energy of a given set S of prime
numbers in a dyadic interval [N, 2N). Here additive energy stands
for the number of solutions to x + y = z + w
with the
variables restricted to lie in the set S.
This bound together with an argument of Hegyvari and Hennecart shows that
when the sequence of primes is coloured with K colours, K an integer
at least 1, then every sufficiently large integer is
the sum of no more than (C K log log K) primes, all of the same colour.
This last bound is optimal up to the value of the
constant C.