Coloring classes of $\{1,2,3,\dots,n\}$ I'm trying to prove the following statement

There is an integer $n_0$ such that for any $n\ge n_0$, in every $9$-coloring of $\{1,2,3,\dots,n\}$, one of the $9$ color classes contains $4$ integers $a,b,c,d$ such that $a+b+c=d$.

I thought about taking $\{1,2,\dots,n\}$ to be vertices of a graph where edges connect vertices of the same color. Then I tried to use Ramsey theorem, but to no avail. I thought also about Schur theorem and tried to find some generalization of it, but got nothing.
How should one prove the statement?
Please help, thanks!

Isn't this a direct consequence of Rado's single equation theorem?
(taken from the book Ramsey Theory over the Integers by B. Landmann and A. Robertson)

where $c_1=c_2=c_3=1$, $c_4=-1$ for your statement and we can take e.g., $D=\{c_1, c_4\}$.  By "regular", it means your statement is true for all $r$-colorings with $r\ge1$.
