## 580 -Partition calculus (6)

April 24, 2009

1. The ${\mbox{Erd\H os}}$-Rado theorem

Large homogeneous sets (of size ${\omega_1}$ or larger) can be ensured, at the cost of starting with a larger domain. The following famous result was originally shown by ${\mbox{Erd\H os}}$ and Rado using tree arguments (with ${\kappa+1}$ lowered to ${\kappa}$ in the conclusion). We give instead an elementary substructures argument due to Baumgartner, Hajnal and ${\mbox{Todor\v cevi\'c},}$ which proves the stronger version. For ${\kappa}$ a cardinal let ${2^{<\kappa}=\sup_{\lambda<\kappa}2^\lambda.}$

Theorem 1 (${\mbox{Erd\H os}}$-Rado) Let ${\kappa}$ be a regular cardinal and let ${\lambda=(2^{<\kappa})^+.}$ Then

$\displaystyle \lambda\rightarrow(\kappa+1)^2_\mu$

for all ${\mu<\kappa.}$