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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issn = {0065-1036},
author = {Szemer{é}di, E.},
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journal = {Acta Arith.},
zblnumber = {0303.10056},
volume = {27},
mrnumber = {0369312},
fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
mrreviewer = {S. L. G. Choi},
title = {On sets of integers containing no {$k$} elements in arithmetic progression},
year = {1975},
pages = {199--245},
} -
[vdWaerden27] B. L. van der Waerden, "Beweis einer baudetschen vermutung," Nieuw. Arch. Wisk., vol. 15, pp. 212-216, 1927.
@ARTICLE{vdWaerden27, zblnumber = {53.0073.12},
volume = {15},
author = {van~der Waerden, B. L.},
title = {Beweis einer baudetschen vermutung},
year = {1927},
pages = {212--216},
journal = {Nieuw. Arch. Wisk.},
} -
[Vinh14]
L. A. Vinh, "Monochromatic sum and product in $\Bbb{Z}/m\Bbb{Z}$," J. Number Theory, vol. 143, pp. 162-169, 2014.
@ARTICLE{Vinh14, mrkey = {3227340},
issn = {0022-314X},
author = {Vinh, Le Anh},
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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},
}