Partition relations for Polish spaces and the Halpern-Läuchli theorem


题目:Partition relations for Polish spaces and the  Halpern-Läuchli theorem

报告人:Chris Lambie-Hanson捷克科学院数学研究所



摘要:The Halpern-Läuchli theorem is an important Ramsey theoretic statement about colorings of finite products of trees. Motivated by the desire to find simple proofs for the Halpern-Läuchli theorem and its variants, we   investigate a family of partition relations for finite products of perfect Polish spaces. The simplest of these can be stated as a question in the following way: given a finite coloring $c$ of the points in some finite-dimensional Euclidean space $\mathbb{R}^n$, must there exist somewhere dense subsets $X_1, X_2, \ldots, X_n$ of $\mathbb{R}$ such that $c$ is constant on the product $X_1 \times X_2 \times \ldots \times X_n$? In most cases, the answer ends up being independent of the axioms of set theory and is tightly connected with the question of the value of the continuum. This is joint work with Andy Zucker.

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