MA and its effect on Tree Partitions

(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). \omega_1 \rightarrow (\omega_1, \omega+1)^2

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

Theorem(Hajnal). Under CH, \omega_1 \not\rightarrow (\omega_1, \omega+2)^2
Theorem (Todorcevic). Under PFA, for any countable ordinal \alpha , \omega_1 \rightarrow (\omega_1, \alpha)^2

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 \rightarrow (Nonspecial Tree, \omega+1)^2

This is the analogue or the Erdos-Rado theorem.

Recall that a tree T is nonspecial if T \rightarrow (\omega)^1_\omega , which means that any countable partition T 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 T a tree with no uncountable chains and \vert T \vert = 2^{\aleph_0} we have T \not\rightarrow (T, \omega+2)^2 .

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