Friday 12 September Fields Institute, Room 210, 13:30-15:00

Speaker: Konstantinos Tyros.

Title: A disjoint union theorem for trees.

Abstract: In this talk we will present an infinitary disjoint union theorem for level products of trees. An easy consequence of the dual Ramsey theorem due to T.J. Carlson and S.G. Simpson is that for every Suslin measurable finite coloring of the power set of the natural numbers, there exists a sequence $(X_n)_{n\in\mathbb{N}}$ of disjoint non-empty subsets of $\mathbb{N}$ such that the set

\[\Big\{\bigcup_{n\in Y}X_n:\; Y\;\text{non-empty subset of }\mathbb{N}\Big\}\]

is monochromatic. The result that we will present is of this sort, where the underline structure is the level product of a finite sequence of uniquely rooted and finitely branching trees with no maximal nodes of height $\omega$ instead of the natural numbers.

As it is required by the proof of the above result, we develop an analogue of the infinite dimensional version of the Hales–Jewett Theorem for maps defined on a level product of trees, which we will also present, if time permits.

\[\Big\{\bigcup_{n\in Y}X_n:\; Y\;\text{non-empty subset of }\mathbb{N}\Big\}\]

is monochromatic. The result that we will present is of this sort, where the underline structure is the level product of a finite sequence of uniquely rooted and finitely branching trees with no maximal nodes of height $\omega$ instead of the natural numbers.

As it is required by the proof of the above result, we develop an analogue of the infinite dimensional version of the Hales–Jewett Theorem for maps defined on a level product of trees, which we will also present, if time permits.