Partition regularity of the three-term reciprocal pattern

A finite coloring of Q\mathbb{Q} is a coloring using finitely many colors. A pattern is partition regular over Q\mathbb{Q} if every finite coloring of Q\mathbb{Q} admits parameters making all entries of the pattern the same color.

Three-term reciprocal-pattern conjecture. The pattern

{x, x+d, x+1d}\left\{x,\ x+d,\ x+\frac{1}{d}\right\}

is partition regular over Q\mathbb{Q}.

This is weaker than the reciprocal parallelogram conjecture, since it omits x+d+1/dx+d+1/d, but the paper states that even this weaker pattern is not known.

Sources & referencesView supporting material

Primary source

Ryan Alweiss, “Monochromatic Sums and Products over Q”, arXiv:2307.08901 (2026).

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