43,164 research outputs found
Ramsey numbers for partially-ordered sets
We present a refinement of Ramsey numbers by considering graphs with a
partial ordering on their vertices. This is a natural extension of the ordered
Ramsey numbers. We formalize situations in which we can use arbitrary families
of partially-ordered sets to form host graphs for Ramsey problems. We explore
connections to well studied Tur\'an-type problems in partially-ordered sets,
particularly those in the Boolean lattice. We find a strong difference between
Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the
partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl
Boolean algebras and Lubell functions
Let denote the power set of . A collection
\B\subset 2^{[n]} forms a -dimensional {\em Boolean algebra} if there
exist pairwise disjoint sets , all non-empty
with perhaps the exception of , so that \B={X_0\cup \bigcup_{i\in I}
X_i\colon I\subseteq [d]}. Let be the maximum cardinality of a family
\F\subset 2^X that does not contain a -dimensional Boolean algebra.
Gunderson, R\"odl, and Sidorenko proved that where .
In this paper, we use the Lubell function as a new measurement for large
families instead of cardinality. The Lubell value of a family of sets \F with
\F\subseteq \tsupn is defined by h_n(\F):=\sum_{F\in \F}1/{{n\choose |F|}}.
We prove the following Tur\'an type theorem. If \F\subseteq 2^{[n]} contains
no -dimensional Boolean algebra, then h_n(\F)\leq 2(n+1)^{1-2^{1-d}} for
sufficiently large . This results implies , where is an absolute constant independent of and . As a
consequence, we improve several Ramsey-type bounds on Boolean algebras. We also
prove a canonical Ramsey theorem for Boolean algebras.Comment: 10 page
The difference between the metric dimension and the determining number of a graph
We study the maximum value of the difference between the metric dimension and the determining number of a graph as a function of its order. We develop a technique that uses functions related to locating-dominating sets to obtain lower and upper bounds on that maximum, and exact computations when restricting to some specific families of graphs. Our approach requires very diverse tools and connections with well-known objects in graph theory; among them: a classical result in graph domination by Ore, a Ramsey-type result by Erdős and Szekeres, a polynomial time algorithm to compute distinguishing sets and determining sets of twin-free graphs, k-dominating sets, and matchings
Unprovability and phase transitions in Ramsey theory
The first mathematically interesting, first-order arithmetical example of incompleteness was given in the late seventies and is know as the Paris-Harrington principle. It is a strengthened form of the finite Ramsey theorem which can not be proved, nor refuted in Peano Arithmetic. In this dissertation we investigate several other unprovable statements of Ramseyan nature and determine the threshold functions for the related phase transitions.
Chapter 1 sketches out the historical development of unprovability and phase transitions, and offers a little information on Ramsey theory. In addition, it introduces the necessary mathematical background by giving definitions and some useful lemmas.
Chapter 2 deals with the pigeonhole principle, presumably the most well-known, finite instance of the Ramsey theorem. Although straightforward in itself, the principle gives rise to unprovable statements. We investigate the related phase transitions and determine the threshold functions.
Chapter 3 explores a phase transition related to the so-called infinite subsequence principle, which is another instance of Ramsey’s theorem.
Chapter 4 considers the Ramsey theorem without restrictions on the dimensions and colours. First, generalisations of results on partitioning α-large sets are proved, as they are needed later. Second, we show that an iteration of a finite version of the Ramsey theorem leads to unprovability.
Chapter 5 investigates the template “thin implies Ramsey”, of which one of the theorems of Nash-Williams is an example. After proving a more universal instance, we study the strength of the original Nash-Williams theorem. We conclude this chapter by presenting an unprovable statement related to Schreier families.
Chapter 6 is intended as a vast introduction to the Atlas of prefixed polynomial equations. We begin with the necessary definitions, present some specific members of the Atlas, discuss several issues and give technical details
The Ramsey property implies no mad families
We show that if all collections of infinite subsets of have the Ramsey
property, then there are no infinite maximal almost disjoint (mad) families.
This solves a long-standing problem going back to Mathias \cite{mathias}. The
proof exploits an idea which has its natural roots in ergodic theory,
topological dynamics, and invariant descriptive set theory: We use that a
certain function associated to a purported mad family is invariant under the
equivalence relation , and thus is constant on a "large" set. Furthermore
we announce a number of additional results about mad families relative to more
complicated Borel ideals.Comment: 10 pages; fixed a mistake in Theorem 4.
Ramseyan ultrafilters
We investigate families of partitions of omega which are related to special
coideals, so-called happy families, and give a dual form of Ramsey ultrafilters
in terms of partitions. The combinatorial properties of these
partition-ultrafilters, which we call Ramseyan ultrafilters, are similar to
those of Ramsey ultrafilters. For example it will be shown that dual Mathias
forcing restricted to a Ramseyan ultrafilter has the same features as Mathias
forcing restricted to a Ramsey ultrafilter. Further we introduce an ordering on
the set of partition-filters and consider the dual form of some cardinal
characteristics of the continuum
Rainbow Ramsey problems for the Boolean lattice
We address the following rainbow Ramsey problem: For posets what is the
smallest number such that any coloring of the elements of the Boolean
lattice either admits a monochromatic copy of or a rainbow copy of
. We consider both weak and strong (non-induced and induced) versions of
this problem. We also investigate related problems on (partial) -colorings
of that do not admit rainbow antichains of size
Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions
Two well studied Ramsey-theoretic problems consider subsets of the natural
numbers which either contain no three elements in arithmetic progression, or in
geometric progression. We study generalizations of this problem, by varying the
kinds of progressions to be avoided and the metrics used to evaluate the
density of the resulting subsets. One can view a 3-term arithmetic progression
as a sequence , where , a nonzero
integer. Thus avoiding three-term arithmetic progressions is equivalent to
containing no three elements of the form with , the set of integer translations. One can similarly
construct related progressions using different families of functions. We
investigate several such families, including geometric progressions ( with a natural number) and exponential progressions ().
Progression-free sets are often constructed "greedily," including every
number so long as it is not in progression with any of the previous elements.
Rankin characterized the greedy geometric-progression-free set in terms of the
greedy arithmetic set. We characterize the greedy exponential set and prove
that it has asymptotic density 1, and then discuss how the optimality of the
greedy set depends on the family of functions used to define progressions.
Traditionally, the size of a progression-free set is measured using the (upper)
asymptotic density, however we consider several different notions of density,
including the uniform and exponential densities.Comment: Version 1.0, 13 page
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