5 research outputs found
Symmetries and Ramsey properties of trees
AbstractIn this paper we show the extent to which a finite tree of fixed height is a Ramsey object in the class of trees of the same height can be measured by its symmetry group
Operators on the Stopping Time Space
Let be the stopping time space and be the Baire-1
elements of the second dual of . To each element in the space
we associate a positive Borel measure on
the Cantor set. We use the measures to characterize the operators ,
defined on a space with an unconditional basis, which preserve a copy of
. In particular, we show that preserves a copy of if and only if
the set is non separable as a
subset of .Comment: 19 page
Dense subsets of products of finite trees
We prove a "uniform" version of the finite density Halpern-L\"{a}uchli
Theorem. Specifically, we say that a tree is homogeneous if it is uniquely
rooted and there is an integer , called the branching number of ,
such that every has exactly immediate successors. We show the
following.
For every integer , every with for all , every integer k\meg 1 and every real
there exists an integer with the following property. If
are homogeneous trees such that the branching number of
is for all , is a finite subset of of
cardinality at least and is a subset of the level product of
satisfying for every , then there
exist strong subtrees of of height and with
common level set such that the level product of is contained in
. The least integer with this property will be denoted by
.
The main point is that the result is independent of the position of the
finite set . The proof is based on a density increment strategy and gives
explicit upper bounds for the numbers .Comment: 36 pages, no figures; International Mathematics Research Notices, to
appea
A density version of the Carlson--Simpson theorem
We prove a density version of the Carlson--Simpson Theorem. Specifically we
show the following.
For every integer and every set of words over satisfying
there exist a word
over and a sequence of left variable words over such that the
set is contained in .
While the result is infinite-dimensional its proof is based on an appropriate
finite and quantitative version, also obtained in the paper.Comment: 73 pages, no figure