Show that for every $k$, there exists a finite integer $n=n(k)$ such that for every coloring of {$1,2,...,n$} by $k$ colors, there are distinct $a,b,c,d$ of the same color satisfying $abc=d$.
I'm considering a method similar to the proof for Schur's Theorem, but I'm not sure how to proceed since we have $abc = d$.