I am teaching a graduate-level introduction to set theory course this semester, and we got to spend some time with one of my favorite set-theory facts: the Continuum Hypothesis holds if and only if there is a partition of \(\mathbb{R}\) into countably many pieces such that no piece contains four distinct numbers satisfying \(x+y = u+v\). For some reason, I prefer the contrapositive formulation: CH fails if and only if for any coloring of \(\mathbb{R}\)with countably many colors, we can find a non-trivial monochromatic solution to the equation \(x+y=u+v\) .
I do not know the original source for this, as I learned it from reading through the wonderful resource Problems and Theorems in Classical Set Theory by Péter Komjáth and Vilmos Totik. Attached to this post is a short writeup that I produced for my students.