10,366 research outputs found
Remarks on a Ramsey theory for trees
Extending Furstenberg's ergodic theoretic proof for Szemer\'edi's theorem on
arithmetic progressions, Furstenberg and Weiss (2003) proved the following
qualitative result. For every d and k, there exists an integer N such that no
matter how we color the vertices of a complete binary tree T_N of depth N with
k colors, we can find a monochromatic replica of T_d in T_N such that (1) all
vertices at the same level in T_d are mapped into vertices at the same level in
T_N; (2) if a vertex x of T_d is mapped into a vertex y in T_N, then the two
children of x are mapped into descendants of the the two children of y in T_N,
respectively; and 3 the levels occupied by this replica form an arithmetic
progression. This result and its density versions imply van der Waerden's and
Szemer\'edi's theorems, and laid the foundations of a new Ramsey theory for
trees.
Using simple counting arguments and a randomized coloring algorithm called
random split, we prove the following related result. Let N=N(d,k) denote the
smallest positive integer such that no matter how we color the vertices of a
complete binary tree T_N of depth N with k colors, we can find a monochromatic
replica of T_d in T_N which satisfies properties (1) and (2) above. Then we
have N(d,k)=\Theta(dk\log k). We also prove a density version of this result,
which, combined with Szemer\'edi's theorem, provides a very short combinatorial
proof of a quantitative version of the Furstenberg-Weiss theorem.Comment: 10 pages 1 figur
Strolling through Paradise
With each of the usual tree forcings I (e.g., I = Sacks forcing S, Laver
forcing L, Miller forcing M, Mathias forcing R, etc.) we associate a
sigma--ideal i^0 on the reals as follows: A \in i^0 iff for all T \in I there
is S \leq T (i.e. S is stronger than T or, equivalently, S is a subtree of T)
such that A \cap [S] = \emptyset, where [S] denotes the set of branches through
S. So, s^0 is the ideal of Marczewski null sets, r^0 is the ideal of Ramsey
null sets (nowhere Ramsey sets) etc.
We show (in ZFC) that whenever i^0, j^0 are two such ideals, then i^0 \sem
j^0 \neq \emptyset. E.g., for I=S and J=R this gives a Marczewski null set
which is not Ramsey, extending earlier partial results by Aniszczyk,
Frankiewicz, Plewik, Brown and Corazza and answering a question of the latter.
In case I=M and J=L this gives a Miller null set which is not Laver null; this
answers a question addressed by Spinas.
We also investigate the question which pairs of the ideals considered are
orthogonal and which are not.
Furthermore we include Mycielski's ideal P_2 in our discussion
Directed Ramsey number for trees
In this paper, we study Ramsey-type problems for directed graphs. We first
consider the -colour oriented Ramsey number of , denoted by
, which is the least for which every
-edge-coloured tournament on vertices contains a monochromatic copy of
. We prove that for any oriented
tree . This is a generalisation of a similar result for directed paths by
Chv\'atal and by Gy\'arf\'as and Lehel, and answers a question of Yuster. In
general, it is tight up to a constant factor.
We also consider the -colour directed Ramsey number
of , which is defined as above, but, instead
of colouring tournaments, we colour the complete directed graph of order .
Here we show that for any
oriented tree , which is again tight up to a constant factor, and it
generalises a result by Williamson and by Gy\'arf\'as and Lehel who determined
the -colour directed Ramsey number of directed paths.Comment: 27 pages, 14 figure
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
Induced Ramsey-type theorems
We present a unified approach to proving Ramsey-type theorems for graphs with
a forbidden induced subgraph which can be used to extend and improve the
earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham,
and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by
Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's
regularity lemma, thereby giving much better bounds. The same approach can be
also used to show that pseudo-random graphs have strong induced Ramsey
properties. This leads to explicit constructions for upper bounds on various
induced Ramsey numbers.Comment: 30 page
Dimension and cut vertices: an application of Ramsey theory
Motivated by quite recent research involving the relationship between the
dimension of a poset and graph-theoretic properties of its cover graph, we show
that for every , if is a poset and the dimension of a subposet
of is at most whenever the cover graph of is a block of the cover
graph of , then the dimension of is at most . We also construct
examples which show that this inequality is best possible. We consider the
proof of the upper bound to be fairly elegant and relatively compact. However,
we know of no simple proof for the lower bound, and our argument requires a
powerful tool known as the Product Ramsey Theorem. As a consequence, our
constructions involve posets of enormous size.Comment: Final published version with updated reference
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