If ${\mathbb{R}}^2$ is finitely coloured then must there exist some colour class which contains the vertices of a rectangle of every area?

AMS 5 51 · solved (literature)

theorem erdos_189 :
    answer(False) ↔ Erdos189For
      (fun a b c d ↦
        line[ℝ, a, b].direction ⟂ line[ℝ, b, c].direction ∧
        line[ℝ, b, c].direction ⟂ line[ℝ, c, d].direction ∧
        line[ℝ, c, d].direction ⟂ line[ℝ, d, a].direction)
      (fun a b c d ↦ dist a b * dist b c)