Erdős problem 484

Prove that there exists an absolute constant $c>0$ such that, whenever $\{1,\ldots,N\}$ is $k$-coloured (and $N$ is large enough depending on $k$) then there are at least $cN$ many integers in $\{1,\ldots,N\}$ which are representable as a monochromatic sum (that is, $a+b$ where $a,b\in \{1,\ldots,N\}$ are in the same colour class and $a\neq b$).

A conjecture of Roth. Solved by Erdős, Sárközy, and Sós [ESS89], who in fact prove that there are at least $\frac{N}{2}-O(N^{1-1/2^{k+1}})$ many even numbers which are of this form.