Erdős-Rado theorem

Repeated exponentiation for cardinals is denoted ${\exp }_i(\kappa )$, where . It is defined by:

$${\exp }_0(\kappa )=\kappa$$

and

$${\exp }_{i+1}(\kappa )=2^{{\exp }_i(\kappa )}$$

The Erdős-Rado theorem states that:

$${\exp }_i{(\kappa )}^{+}\to {({\kappa }^{+})}_{\kappa }^{i+1}$$

That is, if $f:{[{\exp }_i{(\kappa )}^{+}]}^{i+1}\to \kappa$ then there is a homogeneous set of size ${\kappa }^{+}$.

As special cases, ${(2^{\kappa })}^{+}\to {({\kappa }^{+})}_{\kappa }^2$ and ${(2^{{\mathrm{\aleph }}_0})}^{+}\to {({\mathrm{\aleph }}_1)}_{{\mathrm{\aleph }}_0}^2$.

Title	Erdős-Rado theorem
Canonical name	ErdHosRadoTheorem
Date of creation	2013-03-22 12:59:30
Last modified on	2013-03-22 12:59:30
Owner	Henry (455)
Last modified by	Henry (455)
Numerical id	9
Author	Henry (455)
Entry type	Theorem
Classification	msc 05D10
Classification	msc 03E05
Related topic	arrowsrelation
Related topic	ArrowsRelation