Related papers: Partition regularity of infinite parallelepiped se…
In this paper, we provide versions of Van der Waerden's theorem and Rado's theorem for finite colorings of IP-sets and k-IP-sets. Here, by an IP-set we mean a set of integers that contains all finite sums of an infinite subset of N, and we…
A system of homogeneous linear equations with integer coefficients is partition regular if, whenever the natural numbers are finitely coloured, the system has a monochromatic solution. The Finite Sums theorem provided the first example of…
Let $X$ be an uncountable Polish space and let $\mathcal{H}$ be the Hindman ideal, that is, the family of all $S\subseteq \omega$ which are not $IP$-sets. For each sequence $x=(x_n)_{n \in \omega}$ taking values in $X$, let…
A partial semigroup is a set with restricted binary operation. In this work we will extend a result due to V. Bergelson and N. Hindman concerning the rich structure presented in the product space of semigroups to partial semigroup. An…
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…
A finite or infinite matrix $A$ is image partition regular provided that whenever $\mathbb{N}$ is finitely colored, there must be some $\overset{\rightarrow}{x}$ with entries from $\mathbb{N}$ such that all entries of $A…
An infinite integer matrix A is called image partition regular if, whenever the natural numbers are finitely coloured, there is an integer vector x such that Ax is monochromatic. Given an image partition regular matrix A, can we also insist…
A system of linear equations with integer coefficients is partition regular over a subset S of the reals if, whenever S\{0} is finitely coloured, there is a solution to the system contained in one colour class. It has been known for some…
A finite or infinite matrix $A$ is image partition regular provided that whenever $\mathbb N$ is finitely colored, there must be some $\vec{x}$ with entries from $\mathbb N$ such that all entries of $A\vec{x}$ are in some color class. In…
For an infinite set M of natural numbers, let FS(M) be the set of all nonzero finite sums of distinct numbers in M. An IP set is any set of the form FS(M). Let p_n denote the n-th prime number for each $n \ge 1$. A de Polignac number is any…
A finite or infinite matrix $A$ is image partition regular provided that whenever $\mathbb N$ is finitely colored, there must be some $\vec{x}$ with entries from $\mathbb N$ such that all entries of $A\vec{x}$ are in some color class. In…
We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of $\mathbb{N}$ where each integer is included independently with probability…
Hindman's theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. Ramsey algebras are structures that satisfy an analogue of Hindman's Theorem. This paper introduces Ramsey algebras and…
In this paper we study some additive properties of subsets of the set $\nats$ of positive integers: A subset $A$ of $\nats$ is called {\it $k$-summable} (where $k\in\ben$) if $A$ contains $\textstyle \big{\sum_{n\in F}x_n | \emp\neq…
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 Ax=0. Many of the classical theorems of Ramsey Theory may naturally…
A subset $A$ of $\nats$ is called an IP-set if $A$ contains all finite sums of distinct terms of some infinite sequence $(x_n)_{n\in \nats} $ of natural numbers. Central sets, first introduced by Furstenberg using notions from topological…
Hindman's finite sums theorem states that in any finite coloring of the naturals, there is an infinite sequence all of whose finite subset sums are the same color. In 1979, Hindman showed that there is a finite coloring of the naturals so…
Let $T_X$ be the semigroup of all non-invertible transformations on an arbitrary set $X$. It is known that $T_X$ is a regular semigroup. The principal right(left) ideals of a regular semigroup $S$ with partial left(right) translations as…
Hindman's Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue of Hindman's Theorem. It is an open…
$A$ set is called $IP$-set in a semigroup $\left(S,\cdot \right)$ if it contains finite products of a sequence. A set that intersects with all $IP$-sets is called $IP^\star$-set. It is a well known and established result by Bergelson and…