$100

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 $A$. It is known that it cannot be true that
\[\lvert A\cap \{1,\ldots,N\}\rvert \ll (\log N)^2\]
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.

The stated bounds are due to Burr and Erdős [BuEr85].