Erdős–Rado theorem


In partition calculus, part of combinatorial set theory, a branch of mathematics, the Erdős–Rado theorem is a basic result extending Ramsey's theorem to uncountable sets.

Statement of the theorem

If r ≥ 0 is finite and κ is an infinite cardinal, then
where exp0 = κ and inductively expr+1=2expr. This is sharp in the sense that expr+ cannot be replaced by expr on the left hand side.
The above partition symbol describes the following statement. If f is a coloring of the r+1-element subsets of a set of cardinality expr+, in κ many colors, then there is a homogeneous set of cardinality κ+.