Is it true that for any $k \geq 2$, there is an integer $n=n(k)$ such that for any $k$-coloring of $\{1,...,n\},$ the equation $xy=z$ has a monochromatic solution?
2026-03-25 03:07:13.1774408033
monochromatic solution to $xy=z$
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Some hints :
This is a simple modification of Schur's theorem via an exponential.
This document from EPFL on page 35 contains a proof of Schur's theorem which you can adapt to your problem.