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Matematicheskie Zametki, 2010, Volume 88, Issue 4, Pages 625–634
DOI: https://doi.org/10.4213/mzm6581
(Mi mzm6581)
 

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
References:
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
English version:
Mathematical Notes, 2010, Volume 88, Issue 4, Pages 603–611
DOI: https://doi.org/10.1134/S0001434610090336
Bibliographic databases:
Document Type: Article
UDC: 514.7
Language: Russian
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
Citation in format AMSBIB
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  • https://www.mathnet.ru/eng/mzm6581
  • https://doi.org/10.4213/mzm6581
  • https://www.mathnet.ru/eng/mzm/v88/i4/p625
  • This publication is cited in the following 13 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математические заметки Mathematical Notes
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