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This article is cited in 13 scientific papers (total in 13 papers)
On Monochromatic Solutions of Some Nonlinear Equations in $\mathbb Z/p\mathbb Z$
I. D. Shkredov M. V. Lomonosov Moscow State University
Abstract:
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, Fourier transform, trigonometric sum, Cauchy–Bunyakovskii inequality.
Received: 22.12.2009
Citation:
I. D. Shkredov, “On Monochromatic Solutions of Some Nonlinear Equations in $\mathbb Z/p\mathbb Z$”, Mat. Zametki, 88:4 (2010), 625–634; Math. Notes, 88:4 (2010), 603–611
Linking options:
https://www.mathnet.ru/eng/mzm6581https://doi.org/10.4213/mzm6581 https://www.mathnet.ru/eng/mzm/v88/i4/p625
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Abstract page: | 563 | Full-text PDF : | 118 | References: | 82 | First page: | 18 |
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