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
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