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.
Hindman,Monochromatic sums equal to products inN, Integers 11A(2011), Article 10
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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.