Related papers: Combined exponential patterns in multiplicative $I…
In [On $IP^{\star}$sets and central sets, Combinatorica, 14 (1994) 269-277], N. Hindman and V.Bergelson proved additive $IP^{\star}$-sets contain finite sums and finite products of a single sequence. An analogous study was made by A. Sisto…
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…
In their proof of the IP Szemer\'edi theorem, a far reaching extension of the classic theorem of Szemer\'edi on arithmetic progressions, Furstenberg and Katznelson introduced an important class of additively large sets called…
It is known that for an IP${^\star}$ set $A$ in $\mathbb{N}$ and a sequence $< x_{n}>_{n=1}^{\infty}$ there exists a sum subsystem $< y_{n}>_{n=1}^{\infty}$ of $< x_{n}>_{n=1}^{\infty}$ such that $FS(< y_n>_{n=1}^\infty)\cup FP(<…
$A$ 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…
A number $m$ is said to be a $\textit{de Polignac number}$, if infinitely many pairs of consecutive primes exist, such that $m$ can be written as the difference of those consecutive prime numbers. Recently in [ W. D. Banks: Consecutive…
A subset $A$ of $\mathbb{N}$ is called an IP-set if $A$ contains all finite sums of distinct terms of some infinite sequence $(x_n)_{n\in \mathbb{N}} $ of natural numbers. Central sets, first introduced by Furstenberg using notions from…
V. Bergelson and N. Hindman proved that IP$^{\star}$- sets contain all possible finite sums and products of a sum subsystem of any sequence in $\mathbb{N}$. In a recent work the second author of this article has proved that a stronger…
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…
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…
In "IP-sets and polynomial recurrence", Bergelson, Furstenberg, and McCutcheon established the following far reaching extension of Khintchine's recurrence theorem: For any invertible probability preserving system $(X,\mathcal A,\mu,T)$, any…
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…
Using ultrafilter techniques we show that in any partition of $\mathbb{N}$ into 2 cells there is one cell containing infinitely many exponential triples, i.e. triples of the kind $a,b,a^b$ (with $a,b>1$). Also, we will show that any…
A proper infinite parallelepiped (IP) set in a semigroup is an infinite set consisting of a sequence $\myseq{a}$ and its finite sums, or a superset of such a set. Hindman's theorem asserts that the proper IP sets of natural numbers are…
Many natural notions of additive and multiplicative largeness arise from results in Ramsey theory. In this paper, we explain the relationships between these notions for subsets of $\mathbb{N}$ and in more general ring-theoretic structures.…
The \emph{index set} of a computable structure $\mathcal{A}$ is the set of indices for computable copies of $\mathcal{A}$. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary…
We investigate bounds in Ramsey's theorem for relations definable in NIP structures. Applying model-theoretic methods to finitary combinatorics, we generalize a theorem of Bukh and Matousek [B. Bukh, J. Matou\v{s}ek.…
There is a long history of studying Ramsey theory using the algebraic structure of the Stone-\v{C}ech compactification of discrete semigroup. It has been shown that various Ramsey theoretic structures are contained in different algebraic…
This paper analyses the feasible sets structure of general mixed integer linear programs (MIPs) and its relationship with the existence of a finite cardinality test set which can be applied in augmentation algorithms. We derive and…
The study of symmetric structures is a new trend in Ramsey theory. Recently in [7], Di Nasso initiated a systematic study of symmetrization of classical Ramsey theoretical results, and proved a symmetric version of several Ramsey theoretic…