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. It is named after Paul Erdős and Richard Rado. [1] It is sometimes also attributed to Đuro Kurepa who proved it under the additional assumption of the generalised continuum hypothesis, [2] and hence the result is sometimes also referred to as the Erdős–Rado–Kurepa theorem.
If r ≥ 0 is finite and κ is an infinite cardinal, then
where exp0(κ) = κ and inductively expr+1(κ)=2expr(κ). This is sharp in the sense that expr(κ)+ cannot be replaced by expr(κ) 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(κ)+, in κ many colors, then there is a homogeneous set of cardinality κ+ (a set, all whose r+1-element subsets get the same f-value).