Monochromatic solutions to $x + y = z^2$
  Suppose that $\mathbb{N}$ is $2$-coloured. Then there are infinitely many monochromatic solutions to $x + y = z^2$. On the other hand, there is a $3$-colouring of $\mathbb{N}$ with only finitely many monochromatic solutions to this equation. 
Ben Green, Sofia Lindqvist
math.NT