Erdős Problem #53

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 number of integers which are the sum or product of distinct elements of \(A\) is at most \[\exp\left(c \frac{(\log \lvert A\rvert)^2}{\log\log\lvert A\rvert}\right)\] for some constant \(c>0\). (erdosproblems.com multiplies by \(\log\log\lvert A\rvert\); [ErSz83, Theorem 2] divides by it.) See also [52](https://www.erdosproblems.com/52).