SOLVED

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. They also prove that if $k=2$ then there are at least \[\frac{N}{2}-O(\log N)\] many even numbers which are of this form, and that $O(\log N)$ is best possible, since there is a $2$-colouring such that no power of $2$ is representable as a monochromatic sum.

A refinement of this problem appears as Problem 25 on the open problems list of Ben Green.