44 research outputs found
Techniques for approaching the dual Ramsey property in the projective hierarchy
We define the dualizations of objects and concepts which are essential for
investigating the Ramsey property in the first levels of the projective
hierarchy, prove a forcing equivalence theorem for dual Mathias forcing and
dual Laver forcing, and show that the Harrington-Kechris techniques for proving
the Ramsey property from determinacy work in the dualized case as well
Ramsey theorem for trees with successor operation
We prove a general Ramsey theorem for trees with a successor operation. This
theorem is a common generalization of the Carlson-Simpson Theorem and the
Milliken Tree Theorem for regularly branching trees.
Our theorem has a number of applications both in finite and infinite
combinatorics. For example, we give a short proof of the unrestricted
Ne\v{s}et\v{r}il-R\"odl theorem, and we recover the Graham-Rothschild theorem.
Our original motivation came from the study of big Ramsey degrees - various
trees used in the study can be viewed as trees with a successor operation. To
illustrate this, we give a non-forcing proof of a theorem of Zucker on big
Ramsey degrees.Comment: 37 pages, 9 figure