Monochromatic sums and products in $\mathbb N$

Joel Moreira

doi:https://doi.org/10.4007/annals.2017.185.3.10

Abstract

An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair $\{x+y,xy\}$. We answer this question affirmatively in a strong sense by exhibiting a large new class of nonlinear patterns that can be found in a single cell of any finite partition of $\mathbb N$. Our proof involves a correspondence principle that transfers the problem into the language of topological dynamics. As a corollary of our main theorem we obtain partition regularity for new types of equations, such as $x^2-y^2=z$ and $x^2+2y^2-3z^2=w$.

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@ARTICLE{Vinh14, mrkey = {3227340}, issn = {0022-314X}, author = {Vinh, Le Anh}, mrclass = {11B75}, doi = {10.1016/j.jnt.2014.02.004}, journal = {J. Number Theory}, zblnumber = {1296.11017}, volume = {143}, mrnumber = {3227340}, fjournal = {Journal of Number Theory}, mrreviewer = {Donald Jason Gibson}, title = {Monochromatic sum and product in {$\Bbb{Z}/m\Bbb{Z}$}}, year = {2014}, pages = {162--169}, }

Monochromatic sums and products in $\mathbb N$

Abstract

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