Erdős–Rado theorem

In partition calculus, part of combinatorial set theory, which is 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, κ is an infinite cardinal, then

\exp_r(\kappa)^+\longrightarrow(\kappa^+)^{r+1}_\kappa

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).

References