Some Remarks on Goldbach's Problem. By combinatorial means the authors
show the existence of ''thin'' sub-sets of primes, useful for Goldbac
h decompositions. For example, there is a set P' of primes with #{p le
ss-than-or-equal-to x, p is-an-element-of P'} << (ln x)2, such that al
l but O(x(ln x)-A) even integers n less-than-or-equal-to x can be writ
ten as n = p + p' (p prime, p' is-an-element-of P').