## 580 -Partition calculus (5)

April 21, 2009

1. Larger cardinalities

We have seen that ${\omega\rightarrow(\omega)^n_m}$ (Ramsey) and ${\omega\rightarrow[\omega]^n_\omega}$ (${\mbox{Erd\H os}}$-Rado) for any ${n,m<\omega.}$ On the other hand, we also have that ${2^\kappa\not\rightarrow(3)^2_\kappa}$ (${\mbox{Sierpi\'nski}}$) and ${2^\kappa\not\rightarrow(\kappa^+)^2}$ (${\mbox{Erd\H os}}$-Kakutani) for any infinite ${\kappa.}$

Positive results can be obtained for larger cardinals than ${\omega}$ if we relax the requirements in some of the colors. A different extension, the ${\mbox{Erd\H os}}$-Rado theorem, will be discussed later.

Theorem 1 (${\mbox{Erd\H os}}$-Dushnik-Miller) For all infinite cardinals ${\lambda,}$ ${\lambda\rightarrow(\lambda,\omega)^2.}$

This was originally shown by Dushnik and Miller in 1941 for ${\lambda}$ regular, with ${\mbox{Erd\H os}}$ providing the singular case. For ${\lambda}$ regular one can in fact show something stronger:

Theorem 2 (${\mbox{Erd\H os}}$-Rado) Suppose ${\kappa}$ is regular and uncountable. Then
$\displaystyle \kappa\rightarrow_{top}(\mbox{Stationary},\omega+1)^2,$ which means: If ${f:[\kappa]^2\rightarrow2}$ then either there is a stationary ${H\subseteq\kappa}$ that is ${0}$-homogeneous for ${f}$, or else there is a closed subset of ${\kappa}$ of order type ${\omega+1}$ that is ${1}$-homogeneous for ${f}$.

(Above, top stands for “topological.”)