Can one color the positive integers with finitely many colors, so that no two different numbers of the same color add to a square?


Some easy to prove remarks:


1. at least 4 colors are needed, since the sum of any two in $\{386, 2114, 3970, 10430\}$ is square;
2. if $N$ colors suffice for each finite subset, then $N$ colors suffice for all of $\mathbb{N}$;
3. two colors suffice if the squares are replaced by the powers of $2$.



Can the positive integers be colored so that elements of same color never add to a square?
No. See the paper below, which handles more polynomials than just perfect squares.


[On the number of monochromatic solutions of $x+y = z^2$](https://doi.org/10.1017/S0963548305007169). Ayman Khalfalah and Endre Szemerédi. Combinatorics, Probability and Computing (2006) 15, 213–227.


