SOLVED
Let $A$ be a finite set of integers. Is it true that, for every $k$, if $\lvert A\rvert$ is sufficiently large depending on $k$, then there are least $\lvert A\rvert^k$ many integers which are either the sum or product of distinct elements of $A$?
Asked by Erdős and Szemerédi
[ErSz83]. Solved in this form by Chang
[Ch03].
Erdős and Szemerédi proved that there exist arbitrarily large sets $A$ such that the integers which are the sum or product of distinct elements of $A$ is at most
\[\exp\left(c (\log \lvert A\rvert)^2\log\log\lvert A\rvert\right)\]
for some constant $c>0$.
See also [52].
