Geodesic
On Monochromatic Solutions of Some Nonlinear Equations in~$\mathbb Z/p\mathbb Z$
Matematičeskie zametki, Tome 88 (2010) no. 4, pp. 625-634

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Let the set of positive integers be colored in an arbitrary way in finitely many colors (a “finite coloring”). Is it true that, in this case, there are $x,y\in\mathbb Z$ such that $x+y$, $xy$, and $x$ have the same color? This well-known problem of the Ramsey theory is still unsolved. In the present paper, we answer this question in the affirmative in the group $\mathbb Z/p\mathbb Z$, where $p$ is a prime, and obtain an even stronger density result.
Keywords: Ramsey theory, coloring, monochromatic solution, Dirichlet character, trigonometric sum, Cauchy–Bunyakovskii inequality.
Mots-clés : Fourier transform 
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     author = {I. D. Shkredov},
     title = {On {Monochromatic} {Solutions} of {Some} {Nonlinear} {Equations} in~$\mathbb Z/p\mathbb Z$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {625--634},
     publisher = {mathdoc},
     volume = {88},
     number = {4},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_88_4_a13/}
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I. D. Shkredov. On Monochromatic Solutions of Some Nonlinear Equations in~$\mathbb Z/p\mathbb Z$. Matematičeskie zametki, Tome 88 (2010) no. 4, pp. 625-634. http://geodesic.mathdoc.fr/item/MZM_2010_88_4_a13/