The following is an old result of Erdős and Turán (American Mathematical Monthly, 1934):

Given a set of $2^n + 1$ distinct positive integers, all of its two-term sums cannot be composed of the same $n$ primes. (An integer $m$ is said to be composed of primes $p_1, p_2, \dots$ is every prime factor of $m$ is one of those primes. And in two-term sums we exclude sums of two copies of the same integer.)

Let $f(n)$ be the minimum number such that given $f(n)$ distinct positive integers, all its two-term sums cannot be composed of the same $n$ primes. Erdős and Turán conjecture that the upper bound $f(n) \le 2^n + 1$ can be greatly improved. For example, they conjecture that $f(n) = O(n^{1+ \epsilon})$ for every fixed $\epsilon > 0$.

Does anyone know if any progress has been made on this problem since? In particular is there any polynomial upper bound known?

Also I would be interested to know about lower bounds on $f(n)$.