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 Mat. Zametki, 2010, Volume 88, Issue 4, Pages 625–634 (Mi mz6581)

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

DOI: https://doi.org/10.4213/mzm6581

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English version:
Mathematical Notes, 2010, 88:4, 603–611

Bibliographic databases:

UDC: 514.7

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
\Bibitem{Shk10} \by I.~D.~Shkredov \paper On Monochromatic Solutions of Some Nonlinear Equations in~$\mathbb Z/p\mathbb Z$ \jour Mat. Zametki \yr 2010 \vol 88 \issue 4 \pages 625--634 \mathnet{http://mi.mathnet.ru/mz6581} \crossref{https://doi.org/10.4213/mzm6581} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2882224} \transl \jour Math. Notes \yr 2010 \vol 88 \issue 4 \pages 603--611 \crossref{https://doi.org/10.1134/S0001434610090336} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000284073100033} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-78249278442} 

• http://mi.mathnet.ru/eng/mz6581
• https://doi.org/10.4213/mzm6581
• http://mi.mathnet.ru/eng/mz/v88/i4/p625

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