49 research outputs found
New Ramsey Classes from Old
Let C_1 and C_2 be strong amalgamation classes of finite structures, with
disjoint finite signatures sigma and tau. Then C_1 wedge C_2 denotes the class
of all finite (sigma cup tau)-structures whose sigma-reduct is from C_1 and
whose tau-reduct is from C_2. We prove that when C_1 and C_2 are Ramsey, then
C_1 wedge C_2 is also Ramsey. We also discuss variations of this statement, and
give several examples of new Ramsey classes derived from those general results.Comment: 11 pages. In the second version, to be submitted for journal
publication, a number of typos has been removed, and a grant acknowledgement
has been adde
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
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