Sets avoiding a rainbow solution to the generalized Schur equation

A classical result in combinatorial number theory states that the largest subset of $[n]$ avoiding a solution to the equation $x+y=z$ is of size $\lceil n\mathrm{/}2\rceil$. For all integers $k>m$, we prove multicolored extensions of this result where we maximize the sum and product of the sizes of sets $A_1,A_2,\dots ,A_k\subseteq [n]$ avoiding a rainbow solution to the Schur equation $x_1+x_2+\cdots +x_m=x_{m+1}$. Moreover, we determine all the extremal families.