Monochromatic sum and product in Z/mZZ/mZ

Shkredov (2010) [15] showed that if the finite field ZpZp, where p   is a prime, is colored in an arbitrary way in finitely many colors, then there are x,y∈Zpx,y∈Zp such that x+yx+y, xy have the same color. Cilleruelo (2012) [4] extended this result to arbitrary finite fields using Sidon sets. In this short note, we present a graph-theoretic proof of this result. Using the same techniques, we extend this result in the setting of the finite cyclic ring.