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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 $x+y=z^2$?

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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.

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

These slides contain related results and several references.

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