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

${\displaystyle \exp _{r}(\kappa )^{+}\longrightarrow (\kappa ^{+})_{\kappa }^{r+1}}$

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

• Erdős, Paul; Hajnal, András; Máté, Attila; Rado, Richard (1984), Combinatorial set theory: partition relations for cardinals, Studies in Logic and the Foundations of Mathematics, 106, Amsterdam: North-Holland Publishing Co., ISBN 0-444-86157-2, MR 0795592
• Erdős, P.; Rado, R. (1956), "A partition calculus in set theory.", Bull. Amer. Math. Soc., 62: 427–489, doi:10.1090/S0002-9904-1956-10036-0, MR 0081864