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 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.

scholarly journals Ramsey Theory for Layered Semigroups

10.37236/9941 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Jordan Mitchell Barrett
Keyword(s):  
General Framework ◽  
Ramsey Theory ◽  
Partition Theorem ◽  
Finite Sums

We further develop the theory of layered semigroups, as introduced by Farah, Hindman and McLeod, providing a general framework to prove Ramsey statements about such a semigroup $S$. By nonstandard and topological arguments, we show Ramsey statements on $S$ are implied by the existence of coherent sequences in $S$. This framework allows us to formalise and prove many results in Ramsey theory, including Gowers' $\mathrm{FIN}_k$ theorem, the Graham–Rothschild theorem, and Hindman's finite sums theorem. Other highlights include: a simple nonstandard proof of the Graham–Rothschild theorem for strong variable words; a nonstandard proof of Bergelson–Blass–Hindman's partition theorem for located variable words, using a result of Carlson, Hindman and Strauss; and a common generalisation of the latter result and Gowers' theorem, which can be proven in our framework.

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.

scholarly journals Recent results in partition (Ramsey) theory for finite lattices

1981 ◽  
Vol 35 (1-3) ◽  
pp. 185-198 ◽  
Author(s):  
Hans Jürgen Prömel ◽  
Bernd Voigt
Keyword(s):  
Ramsey Theory ◽  
Finite Lattices

On finite sums of periodic functions

2020 ◽  
Vol 27 (2) ◽  
pp. 265-269
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Analytic Function ◽  
Real Line ◽  
Real Analytic Function ◽  
Periodic Functions ◽  
Uniform Limit ◽  
The Real ◽  
Pointwise Limit ◽  
Finite Sums ◽  
Real Analytic

AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

scholarly journals Complete and incomplete Bell polynomials associated with Lah–Bell numbers and polynomials

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Lee-Chae Jang ◽  
Hyunseok Lee ◽  
Han-Young Kim
Keyword(s):  
Bell Polynomials ◽  
Bell Numbers ◽  
Bell Number ◽  
Finite Sums

AbstractThe nth r-extended Lah–Bell number is defined as the number of ways a set with $n+r$ n + r elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah–Bell polynomials and complete r-extended Lah–Bell polynomials respectively as multivariate versions of r-Lah numbers and the r-extended Lah–Bell numbers and to investigate some properties and identities for these polynomials. From these investigations we obtain some expressions for the r-Lah numbers and the r-extended Lah–Bell numbers as finite sums.

scholarly journals Abstract topological Ramsey theory for nets

2016 ◽  
Vol 201 ◽  
pp. 314-329 ◽  
Author(s):  
Vassiliki Farmaki ◽  
Dimitris Karageorgos ◽  
Andreas Koutsogiannis ◽  
Andreas Mitropoulos
Keyword(s):  
Ramsey Theory

Maxima and Minima of Finite Sums

1936 ◽  
Vol 43 (3) ◽  
pp. 164
Author(s):  
Tomlinson Fort
Keyword(s):  
Finite Sums
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