# Monochromatic solution to $x+y=z^2$

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 $$x+y=z^2$$?

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.
Theorem 3 of this paper by Csikvari, Gyarmati, and Sarkozy shows that $$n_0(16)$$ does not exist.