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.}

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