We show that for every 2-coloring of N and every k 2 N, there are infinitely many monochromatic solutions of the system of k2 equations zij = xyj i , 1 ? i, j ? k, where x1, . . . , xk, y1, . . . , yk are distinct positive integers greater than 1. We give similar, but somewhat weaker, results for more than two colors. – Dedicated to the memory of Paul Erd?os.
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