Does anyone know any references/hints for the following problem?


For any $k \geq 1$ there is a threshold, $n\_{0}=n\_{0}(k)$ such that if $n \geq n\_{0}$ then any $k$ -colouring of the first $n$ integers
contains three numbers $x, y, z \in[n]$ from the same colour class giving solution to the $x+y=z^2$?



Monochromatic solution to $x+y=z^2$
A complete answer is given by Green and Lindqvist (<https://arxiv.org/abs/1608.08374>). They show that $n\_0(k)$ does not exist for any $k\geq 3$ but $n\_0(2)$ does exist.


