(*This is the presentation I gave for Stevo Todorcevic’s course Combinatorial Set Theory on Feb 28, 2012. The material comes from Stevo’s 1983 paper “Partition Relations for Partially Ordered Sets”*.)

In partition relations for ordinals, it has been established that:

Theorem (Erdos-Rado).

Later it was shown that this is the best you can do, as the strengthenings are consistent:

Theorem(Hajnal). Under CH,

Theorem (Todorcevic). Under PFA, for any countable ordinal ,

Moving on, we can ask the same questions about non-special trees, which in some way are the tree analogue of “uncountable” or “large”.

Theorem (Todorcevic).Nonspecial Tree (Nonspecial Tree,

This is the analogue or the Erdos-Rado theorem.

Recall that a tree is nonspecial if , which means that any countable partition contains an infinite set. (This is a generalization of uncountable, because for countable sets you can always put one element per colour.)

We will show the following:

Theorem (Todorcevic). Under MA, for a tree with no uncountable chains and we have .