Monochromatic solutions to $x+y=z^2$ in the interval $[N,c{N}^4]$

Green and Lindqvist proved that for any 2-colouring of $\mathbb{N}$, there are in-fi-ni-tely many monochromatic solutions to $x+y=z^2$. In fact, they showed the existence of a monochromatic solution in every interval $[N,c{N}^8]$ with large enough $N$. In this short note we give a different proof for their theorem and prove that a monochromatic solution exists in every interval $[N,{10}^4{N}^4]$ with large enough $N$. A 2-colouring of $[N,(1\mathrm{/}27)N^4]$ avoiding monochromatic solutions to $x+y=z^2$ is also presented, which shows that in ${10}^4{N}^4$ only the constant factor can be reduced.