Freiling theorem ($A_{<{\omega }_1}(\mathbb{R})⟺\neg \mathrm{CH}$)

In ZFC, the Freiling axiom of symmetry is equivalent to failure of the Continuum hypothesis. Under the hypothesis, countable initial segments of a well-order of the reals violate symmetry. When the continuum exceeds ${\aleph }_1$, the union of the countable values on an ${\aleph }_1$-sized set leaves a real outside; avoiding that real's countable value gives the two witnesses.