## 580 -Partition calculus (3)

April 6, 2009

1. Infinitary Jónsson algebras

Once again, assume choice throughout. Last lecture, we showed that ${\kappa\not\rightarrow(\kappa)^{\aleph_0}}$ for any ${\kappa.}$ The results below strengthen this fact in several ways.

Definition 1 Let ${x}$ be a set. A function ${f:[x]^{\aleph_0}\rightarrow x}$ is ${\omega}$-Jónsson for ${x}$ iff for all ${y\subseteq x,}$ if ${|y|=|x|,}$ then ${f''[y]^{\aleph_0}=x.}$

Actually, for ${x=\lambda}$ a cardinal, the examples to follow usually satisfy the stronger requirement that ${f''[y]^\omega=\lambda.}$ In the notation from Definition 16 from last lecture, ${\lambda\not\rightarrow[\lambda]^\omega_\lambda.}$

The following result was originally proved in 1966 with a significantly more elaborate argument. The proof below, from 1976, is due to Galvin and Prikry.

Theorem 2 (Erdös-Hajnal) For any infinite ${x,}$ there is an ${\omega}$-Jónsson function for ${x.}$