For $r ≥ 1$ I need to somehow show that there exists a smallest natural number $S'(r)$, such that every $r$ − coloring of the integers ${1, 2, . . . , N }$, with $N$ ≥ $S'(r)$, will necessarily contain three integers $x$, $y$, $z$, of the same color, such that $z = xy$.
I do know Schur's theorem and I can see the obvious similarity, but I can't seem to connect it with this one.
Any inputs would be really helpful
First, you want to say something to exclude the trivial monochromatic solution $x=y=z=1$, which is always there.
In fact, we can prove this statement with the extra property that $x,y,z$ are powers of $2$ greater than $1$. This follows by applying Schur's Theorem to the powers of 2.
By Schur's Theorem, let $S(r)$ be such that every $r$-coloring of $\{1,2,\ldots,S(r)\}$ contains monochromatic $x$, $y$, and $x+y$. Now let $N=2^{S(r)}$. Consider an $r$-coloring of $\{1,2,3,\ldots,N\}$. Now restrict this coloring to the subset $\{2^k:1\leq k\leq S(r)\}$, and transfer it to a coloring of $\{1,2,\ldots,S(r)\}$ using $2^k\mapsto k$. I leave the rest to you.
In conclusion, we get $S'(r)\leq 2^{S(r)}$ where $S(r)$ is the standard additive Schur number for $r$-colorings.