In dense subsemigroups of ((0,∞), +), the set {a, b, ab, b(a+1)} is monochromatic near zero under any finite coloring, and the pattern x, y, x+y, y/x is partition regular near zero.
Monochromatic sums and quotients in $\mathbb N$
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abstract
We prove partition regularity of the configuration $x,y,x+y,y/x$ in a strong infinitary form that extends Hindman's Theorem. We study the related issue of partition regularity of configurations involving products of a degree one polynomial in $x$ with one in $y$, reducing the general problem to a handful of special cases.
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Monochromatic Sums and Quotients Near Zero
In dense subsemigroups of ((0,∞), +), the set {a, b, ab, b(a+1)} is monochromatic near zero under any finite coloring, and the pattern x, y, x+y, y/x is partition regular near zero.