## Departmental PhD Thesis Exam – Ari Brodsky

Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.

Monday, March 17, 2014
2:00 p.m.
BA6183, 40 St George St.

PhD Candidate: Ari Brodsky

Supervisor: Stevo Todorcevic

Thesis title: A Theory of Stationary Trees and the Balanced Baumgartner-Hajnal-Todorcevic Theorem for Trees

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Building on early work by Stevo Todorcevic, we develop a theory of stationary subtrees of trees of successor-cardinal height. We de fine the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary.

We then use this theory to prove the following partition relation for trees:

Main Theorem. Let  $k$ be any in nite regular cardinal, let $\xi$ be any ordinal such that $2^{|\xi|} < k$ , and let $k$ be any natural number. Then

$non-(2^{< k})-special \ tree \rightarrow (k + \xi)^2_k.$

This is a generalization to trees of the Balanced Baumgartner-Hajnal-Todorcevic Theorem, which we recover by applying the above to the cardinal $(2^{< k})^+$, the simplest example of a non-$(2^{< k})$-special tree.

As a corollary, we obtain a general result for partially ordered sets:

Theorem. Let  $k$ be any in finite regular cardinal, let  $\xi$ be any ordinal such that $2^{|\xi|} < k$ , and let $k$ be any natural number. Let P be a partially ordered set such that $P \rightarrow (2^{<k})^1_{2^{<k}}$. Then

$P \rightarrow (k+\xi)^2_k.$

The thesis is attached here: Brodsky-thesis