Editing Ramsey's theorem (section)

=== Uncountable cardinals ===
{{Main|Partition calculus}}
In terms of the partition calculus, Ramsey's theorem can be stated as <math>\alef_0\rightarrow(\alef_0)^n_k</math> for all finite ''n'' and ''k''. [[Wacław Sierpiński]] showed that the Ramsey theorem does not extend to graphs of size <math>\alef_1</math> by showing that <math>2^{\alef_0}\nrightarrow(\alef_1)^2_2</math>. In particular, the [[continuum hypothesis]] implies that <math>\alef_1\nrightarrow(\alef_1)^2_2</math>. [[Stevo Todorčević]] showed that in fact in [[ZFC]], <math>\alef_1\nrightarrow[\alef_1]^2_{\alef_1}</math>, a much stronger statement than <math>\alef_1\nrightarrow(\alef_1)^2_2</math>. [[Justin T. Moore]] has strengthened this result further. On the positive side, a [[Ramsey cardinal]] is a [[large cardinal]] <math>\kappa</math> axiomatically defined to satisfy the related formula: <math>\kappa\rightarrow(\kappa)^{<\omega}_2</math>. The existence of Ramsey cardinals cannot be proved in ZFC.