A particular case of Schur's problem. There exists a largest integer $s(k)$ such that there exists a partition of $\{1,2,\ldots,s(k)\}$ into $k$ classes, so that the equation $x+y=z$ has no solution in any class.
Schur's estimates are $\dfrac {3^k-1}{2} \leq s(k) < \lfloor k!\textrm{e}\rfloor $, and in fact $s(k) = \dfrac {3^k-1}{2}$ for $k=1,2,3$. Already for $k=4$ the value $\dfrac {3^k-1}{2}$ is too low;
the formula yields $40$, but Baumert computed $s(4) = 44$ (reference [R. K. Guy - Unsolved Problems in Number Theory]).
For our problem, since by Schur's $s(3) < \lfloor 3!\textrm{e}\rfloor = 16 < 40$, it means we need $k\geq 4$. But Schur's $40 = \dfrac {3^4-1}{2} \leq s(4)$ does not yield the answer, so we need exhibit Baumert's example $\{1,3,5,15,17,19,26,28,40,(42),(44)\}$, $\{2,7,8,18,21,24,27,33,37,38,(43)\}$, $\{4,6,13,20,22,23,25,30,32,39,(41)\}$, $\{9,10,11,12,14,16,29,31,34,35,36\}$.