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$$|A\cap \{1,\dots ,N\}|\le c(\log N{)}^2$$for all large $N$ and that there exists a Ramsey $2$-complete $A$ such that for all large $N$$$|A\cap \{1,\dots ,N\}|<(2{\log }_{2}N{)}^3.$$Improve either of these bounds.