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.

Statement of the theorem

If r ≥ 0 is finite and κ is an infinite cardinal, then

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

where exp₀(κ) = κ and inductively exp_r+1(κ)=2^exp_r(κ). This is sharp in the sense that exp_r(κ)⁺ cannot be replaced by exp_r(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 exp_r(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).

Notes

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, vol. 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 (5): 427–489, doi:10.1090/S0002-9904-1956-10036-0, MR 0081864
• Erdős, Paul; Hajnal, András; Rado, Richard (1965), "Partition relations for cardinal numbers", Acta Math. Acad. Sci. Hungar., 16: 93–196, CiteSeerX 10.1.1.299.7622, doi:10.1007/BF01886396, MR 0202613