8,388 research outputs found
Density theorems for bipartite graphs and related Ramsey-type results
In this paper, we present several density-type theorems which show how to
find a copy of a sparse bipartite graph in a graph of positive density. Our
results imply several new bounds for classical problems in graph Ramsey theory
and improve and generalize earlier results of various researchers. The proofs
combine probabilistic arguments with some combinatorial ideas. In addition,
these techniques can be used to study properties of graphs with a forbidden
induced subgraph, edge intersection patterns in topological graphs, and to
obtain several other Ramsey-type statements
A unified approach to Hindman, Ramsey and van der Waerden spaces
For many years, there have been conducting research (e.g. by Bergelson,
Furstenberg, Kojman, Kubi\'{s}, Shelah, Szeptycki, Weiss) into sequentially
compact spaces that are, in a sense, topological counterparts of some
combinatorial theorems, for instance Ramsey's theorem for coloring graphs,
Hindman's finite sums theorem and van der Waerden's arithmetical progressions
theorem. These spaces are defined with the aid of different kinds of
convergences: IP-convergence, R-convergence and ordinary convergence.
The first aim of this paper is to present a unified approach to these various
types of convergences and spaces. Then, using this unified approach, we prove
some general theorems about existence of the considered spaces and show that
all results obtained so far in this subject can be derived from our theorems.
The second aim of this paper is to obtain new results about the specific
types of these spaces. For instance, we construct a Hausdorff Hindman space
that is not an \I_{1/n}-space and a Hausdorff differentially compact space
that is not Hindman. Moreover, we compare Ramsey spaces with other types of
spaces. For instance, we construct a Ramsey space that is not Hindman and a
Hindman space that is not Ramsey.
The last aim of this paper is to provide a characterization that shows when
there exists a space of one considered type that is not of the other kind. This
characterization is expressed in purely combinatorial manner with the aid of
the so-called Kat\v{e}tov order that has been extensively examined for many
years so far.
This paper may interest the general audience of mathematicians as the results
we obtain are on the intersection of topology, combinatorics, set theory and
number theory
Open questions about Ramsey-type statements in reverse mathematics
Ramsey's theorem states that for any coloring of the n-element subsets of N
with finitely many colors, there is an infinite set H such that all n-element
subsets of H have the same color. The strength of consequences of Ramsey's
theorem has been extensively studied in reverse mathematics and under various
reducibilities, namely, computable reducibility and uniform reducibility. Our
understanding of the combinatorics of Ramsey's theorem and its consequences has
been greatly improved over the past decades. In this paper, we state some
questions which naturally arose during this study. The inability to answer
those questions reveals some gaps in our understanding of the combinatorics of
Ramsey's theorem.Comment: 15 page
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
Survey on the Tukey theory of ultrafilters
This article surveys results regarding the Tukey theory of ultrafilters on
countable base sets. The driving forces for this investigation are Isbell's
Problem and the question of how closely related the Rudin-Keisler and Tukey
reducibilities are. We review work on the possible structures of cofinal types
and conditions which guarantee that an ultrafilter is below the Tukey maximum.
The known canonical forms for cofinal maps on ultrafilters are reviewed, as
well as their applications to finding which structures embed into the Tukey
types of ultrafilters. With the addition of some Ramsey theory, fine analyses
of the structures at the bottom of the Tukey hierarchy are made.Comment: 25 page
Ramsey-type theorems for lines in 3-space
We prove geometric Ramsey-type statements on collections of lines in 3-space.
These statements give guarantees on the size of a clique or an independent set
in (hyper)graphs induced by incidence relations between lines, points, and
reguli in 3-space. Among other things, we prove that: (1) The intersection
graph of n lines in R^3 has a clique or independent set of size Omega(n^{1/3}).
(2) Every set of n lines in R^3 has a subset of n^{1/2} lines that are all
stabbed by one line, or a subset of Omega((n/log n)^{1/5}) such that no
6-subset is stabbed by one line. (3) Every set of n lines in general position
in R^3 has a subset of Omega(n^{2/3}) lines that all lie on a regulus, or a
subset of Omega(n^{1/3}) lines such that no 4-subset is contained in a regulus.
The proofs of these statements all follow from geometric incidence bounds --
such as the Guth-Katz bound on point-line incidences in R^3 -- combined with
Tur\'an-type results on independent sets in sparse graphs and hypergraphs.
Although similar Ramsey-type statements can be proved using existing generic
algebraic frameworks, the lower bounds we get are much larger than what can be
obtained with these methods. The proofs directly yield polynomial-time
algorithms for finding subsets of the claimed size.Comment: 18 pages including appendi
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