**1. The -Rado theorem **

Large homogeneous sets (of size or larger) can be ensured, at the cost of starting with a larger domain. The following famous result was originally shown by and Rado using tree arguments (with lowered to in the conclusion). We give instead an elementary substructures argument due to Baumgartner, Hajnal and which proves the stronger version. For a cardinal let

Theorem 1 (-Rado)Let be a regular cardinal and let Then

for all