scholarly journals Extensions of Infinite Partition Regular Systems

10.37236/4568 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Neil Hindman ◽  
Imre Leader ◽  
Dona Strauss
Keyword(s):  
Positive Result ◽  
Ramsey Theory ◽  
Infinite Matrix ◽  
Natural Numbers ◽  
Infinite Systems ◽  
Image Partition ◽  
Algebraic Properties ◽  
New Variables

A finite or infinite matrix $A$ with rational entries (and only finitely many non-zero entries in each row) is called image partition regular if, whenever the natural numbers are finitely coloured, there is a vector $x$, with entries in the natural numbers, such that $Ax$ is monochromatic. Many of the classicial results of Ramsey theory are naturally stated in terms of image partition regularity.Our aim in this paper is to investigate maximality questions for image partition regular matrices.  When is it possible to add rows on to $A$ and remain image partition regular? When can one add rows but `nothing new is produced'? What about adding rows and also new variables? We prove some results about extensions of the most interesting infinite systems, and make several conjectures.Our most surprising positive result is a compatibility result for Milliken-Taylor systems, stating that (in many cases) one may adjoin one Milliken-Taylor system to a translate of another and remain image partition regular. This is in contrast to earlier results, which had suggested a strong inconsistency between different Milliken-Taylor systems. Our main tools for this are some algebraic properties of $\beta {\mathbb N}$, the Stone-Čech compactification of the natural numbers.

Open Problems in Partition Regularity

2003 ◽  
Vol 12 (5-6) ◽  
pp. 571-583 ◽  
Author(s):  
Neil Hindman ◽  
Imre Leader ◽  
Dona Strauss
Keyword(s):  
Ramsey Theory ◽  
Infinite Matrix ◽  
Natural Numbers ◽  
Open Problems ◽  
Finite Case ◽  
Infinite Case

A finite or infinite matrix A with rational entries is called partition regular if, whenever the natural numbers are finitely coloured, there is a monochromatic vector x with . Many of the classical theorems of Ramsey Theory may naturally be interpreted as assertions that particular matrices are partition regular.While in the finite case partition regularity is well understood, very little is known in the infinite case. Our aim in this paper is to present some of the natural and appealing open problems in the area.

scholarly journals On radicals of infinite matrix rings

1969 ◽  
Vol 16 (3) ◽  
pp. 195-203 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Infinite Matrix ◽  
Natural Numbers ◽  
Index Set ◽  
Matrix Rings ◽  
Infinite Set

Let R be a ring and I an infinite set. We denote by M(R) the ring of row finite matrices over I with entries in R. The set I will be omitted from the notation, as the same index set will be used throughout the paper. For convenience it will be assumed that the set of natural numbers is a subset of I.

Rainbow Arithmetic Progressions and Anti-Ramsey Results

2003 ◽  
Vol 12 (5-6) ◽  
pp. 599-620 ◽  
Author(s):  
V Jungic ◽  
J Licht ◽  
M Mahdian ◽  
J Nesetril ◽  
R Radoicic
Keyword(s):  
Arithmetic Progression ◽  
Ramsey Theory ◽  
Arithmetic Progressions ◽  
Natural Numbers ◽  
Colour Class ◽  
Ramsey Type ◽  
General Perspective

The van der Waerden theorem in Ramsey theory states that, for every k and t and sufficiently large N, every k-colouring of [N] contains a monochromatic arithmetic progression of length t. Motivated by this result, Radoičić conjectured that every equinumerous 3-colouring of [3n] contains a 3-term rainbow arithmetic progression, i.e., an arithmetic progression whose terms are coloured with distinct colours. In this paper, we prove that every 3-colouring of the set of natural numbers for which each colour class has density more than 1/6, contains a 3-term rainbow arithmetic progression. We also prove similar results for colourings of . Finally, we give a general perspective on other anti-Ramsey-type problems that can be considered.

On Infinite Systems of Linear Differential Equations

1975 ◽  
Vol 27 (3) ◽  
pp. 691-703 ◽  
Author(s):  
J. P. McClure ◽  
R. Wong
Keyword(s):  
Differential Equations ◽  
Infinite System ◽  
Initial Conditions ◽  
Linear Differential Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Complex Numbers ◽  
Infinite Matrix ◽  
Infinite Systems ◽  
Linear Differential ◽  
Image Position

Let A = [αtj] (i,j = 1, 2, …) be an infinite matrix with complex entries, and let z = (ζj) (j = 1, 2, …) be a sequence of complex numbers. In this paper we wish to investigate the existence, uniqueness and asymptotic behavior of solutions to the infinite system of linear differential equationswith the initial conditions

scholarly journals Universally Image Partition Regularity

10.37236/865 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Dibyendu De ◽  
Ram Krishna Paul
Keyword(s):  
Ramsey Theory ◽  
Schur’S Theorem ◽  
Image Partition ◽  
Finite Sums

Many of the classical results of Ramsey Theory, for example Schur's Theorem, van der Waerden's Theorem, Finite Sums Theorem, are naturally stated in terms of image partition regularity of matrices. Many characterizations are known of image partition regularity over ${\Bbb N}$ and other subsemigroups of $({\Bbb R},+)$. In this paper we introduce a new notion which we call universally image partition regular matrices, which are in fact image partition regular over all semigroups and everywhere. We also prove that such matrices exist in abundance.

scholarly journals Combining Extensions of the Hales-Jewett Theorem with Ramsey Theory in Other Structures

10.37236/7955 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Neil Hindman ◽  
Dona Strauss ◽  
Luca Q. Zamboni
Keyword(s):  
Algebraic Structure ◽  
Free Semigroup ◽  
Ramsey Theory ◽  
Infinite Sequence ◽  
The Other ◽  
Algebraic Proof ◽  
Natural Numbers ◽  
Relevant Variable ◽  
The Given

The Hales-Jewett Theorem states that given any finite nonempty set ${\mathbb A}$ and any finite coloring of the free semigroup $S$ over the alphabet ${\mathbb A}$ there is a variable word over ${\mathbb A}$ all of whose instances are the same color. This theorem has some extensions involving several distinct variables occurring in the variable word. We show that, when combined with a sufficiently well behaved homomorphism,  the relevant variable word simultaneously satisfies a Ramsey-Theoretic conclusion in the other structure. As an example we show that if $\tau$ is the homomorphism from the set of variable words into the natural numbers which associates to each variable word $w$ the number of occurrences of the variable in $w$, then given any finite coloring of $S$ and any infinite sequence of natural numbers, there is a variable word $w$ whose instances are monochromatic and $\tau(w)$ is a sum of distinct members of the given sequence.Our methods rely on the algebraic structure of the Stone-Čech compactification of $S$ and the other semigroups that we consider.  We show for example that if $\tau$ is as in the paragraph above, there is a compact subsemigroup $P$ of $\beta{\mathbb N}$ which contains all of the idempotents of $\beta{\mathbb N}$ such that, given any $p\in P$, any $A\in p$, and any finite coloring of $S$, there is a variable word $w$ whose instances are monochromatic and $\tau(w)\in A$.We end with a new short algebraic proof of an infinitary extension of the Graham-Rothschild Parameter Sets Theorem.

scholarly journals On Zweier generalized difference ideal convergent sequences in a locally convex space defined by Musielak-Orlicz function

2017 ◽  
Vol 35 (2) ◽  
pp. 19 ◽  
Author(s):  
Bipan Hazarika ◽  
Karan Tamanag
Keyword(s):  
Orlicz Function ◽  
Convergent Sequence ◽  
Locally Convex Space ◽  
Sequence Spaces ◽  
Infinite Matrix ◽  
Difference Matrix ◽  
New Class ◽  
Algebraic Properties ◽  
Locally Convex ◽  
Convergent Sequences

Let $\mathbf{M}=(M_k)$ be a Musielak-Orlicz function. In this article, we introduce a new class of ideal convergent sequence spaces defined by Musielak-Orlicz function, using an infinite matrix, and a generalized difference matrix operator $B_{(i)}^{p}$ in locally convex spaces. We investigate some linear topological structures and algebraic properties of these spaces. We obtainsome relations related to these sequence spaces.

Some results in polychromatic Ramsey theory

2007 ◽  
Vol 72 (3) ◽  
pp. 865-896 ◽  
Author(s):  
Uri Abraham ◽  
James Cummings ◽  
Clifford Smyth
Keyword(s):  
Positive Result ◽  
Dual Problem ◽  
Ramsey Theory ◽  
Order Type

Classical Ramsey theory (at least in its simplest form) is concerned with problems of the following kind: given a set X and a colouring of the set [X]n of unordered n-tuples from X, find a subset Y ⊆ X such that all elements of [Y]n get the same colour. Subsets with this property are called monochromatic or homogeneous, and a typical positive result in Ramsey theory has the form that when X is large enough and the number of colours is small enough we can expect to find reasonably large monochromatic sets.Polychromatic Ramsey theory is concerned with a “dual” problem, in which we are given a colouring of [X]n and are looking for subsets Y ⊆ X such that any two distinct elements of [Y]n get different colours. Subsets with this property are called polychromatic or rainbow. Naturally if we are looking for rainbow subsets then our task becomes easier when there are many colours. In particular given an integer k we say that a colouring is k-bounded when each colour is used for at most k many n-tuples.At this point it will be convenient to introduce a compact notation for stating results in polychromatic Ramsey theory. We recall that in classical Ramsey theory we write to mean “every colouring of [κ]n in k colours has a monochromatic set of order type α”. We will write to mean “every k-bounded colouring of [κ]n has a polychromatic set of order type α”. We note that when κ is infinite and k is finite a k-bounded colouring will use exactly κ colours, so we may as well assume that κ is the set of colours used.

scholarly journals Applications of Lacunary Sequences to Develop Fuzzy Sequence Spaces for Ideal Convergence and Orlicz Function

2020 ◽  
Vol 13 (5) ◽  
pp. 1131-1148
Author(s):  
Kuldip Raj ◽  
S. A. Mohiuddine
Keyword(s):  
Maximal Ideal ◽  
Orlicz Function ◽  
Lacunary Sequence ◽  
Sequence Spaces ◽  
Infinite Matrix ◽  
Ideal Convergence ◽  
Lacunary Sequences ◽  
Algebraic Properties

In the present paper, we introduce and study ideal convergence of some fuzzy sequence spaces via lacunary sequence, infinite matrix and Orlicz function. We study some topological and algebraic properties of these spaces. We also make an effort to show that these spaces are normal as well as monotone. Further, it is very interesting to show that if $I$ is not maximal ideal then these spaces are not symmetric.

Image Partition Regularity of Matrices

1993 ◽  
Vol 2 (4) ◽  
pp. 437-463 ◽  
Author(s):  
Neil Hindman ◽  
Imre Leader
Keyword(s):  
General Problem ◽  
Ramsey Theory ◽  
Image Partition ◽  
Positive Integers

Many of the classical results of Ramsey Theory, including those of Hilbert, Schur, and van der Waerden, are naturally stated as instances of the following problem: given a u × ν matrix A with rational entries, is it true, that whenever the set ℕ of positive integers is finitely coloured, there must exist some x∈ℕν such that all entries of Ax are the same colour? While the theorems cited are all consequences of Rado's theorem, the general problem had remained open. We provide here several solutions for the alternate problem, which asks that x∈ℕν. Based on this, we solve the general problem, giving various equivalent characterizations.

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