15 papers
A Uniform Product-Difference Theorem for Dense Subsets of
Sayan Goswami, Chunlin Liu
We establish a uniform product-difference theorem for dense subsets of , which gives an affirmative answer to Problem~2 of Fish and, as consequences, to both parts of…
On difference sets of dense subsets of
Sayan Goswami
In this article, we study the structure of the difference set for subsets of positive upper Banach density. Fish asked in [Proc. Amer. Math. Soc.…
Random Finite Sumsets and Product Sets in Subsets of the Natural Numbers
Sukrit Chakraborty, Sayan Goswami, Sourav Kanti Patra
We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of where…
Moreira's Theorem for Image Partition Regular Matrices
Sayan Goswami
The famous Hindman conjecture says that for any finite coloring of natural numbers, there exists a monochromatic copy of the form In a celebrated article, Moreira g…
Homogeneous Patterns in Ramsey Theory
Sukumar Das Adhikari, Sayan Goswami
In this article, we investigate homogeneous versions of certain nonlinear Ramsey-theoretic results, with three significant applications. As the first application, we prove that for…
Monochromatic Translated Product and Answering Sahasrabudhe's Conjecture
Sayan Goswami
This article resolves two related problems in Ramsey theory on the integers. We show that for any finite coloring of the set of natural numbers, there exist numbers and for…
Exponential Schur and Hindman Theorem in Ramsey Theory
Sayan Goswami, Sourav Kanti Patra
Answering a conjecture of A. Sisto, J. Sahasrabudhe proved the exponential version of the Schur theorem: for every finite coloring of the naturals, there exists a monochromatic cop…
Restricted van der Waerden theorem for nilprogressions
Sayan Goswami
In [Adv. Math., 321 (2017) 269-286], using the theory of ultrafilters, J. H. Johnson Jr., and F. K. Richter proved the nilpotent polynomial Hales-Jewett theorem. Using this result…
Nonlinear Kernel Partition Regularity: Necessary and Sufficient Conditions
Sayan Goswami
A matrix \( A \) is called \emph{kernel partition regular} if, for every finite coloring of the natural numbers \( \mathbb{N} \), there exists a monochromatic solution to the equat…
Dynamical characterization of central sets in adequate partial semigroups
Pintu Debnath, Sayan Goswami, Sourav Kanti Patra
Using the methods from topological dynamics, H. Furstenberg introduced the notions of Central sets and proved the famous Central Sets Theorem which is the simultaneous extension of…
An Explicit Ramsey Bound for de Polignac Numbers
Sayan Goswami
A number is called a \textbf{de Polignac number}( in short) if it can be expressed as the difference of infinitely many pairs of consecutive prime numbers. Fo…
Multidimensional Stronger Central Sets Theorem and its Polynomial Extension
Sayan Goswami, Sourav Kanti Patra
We establish and fully characterize the multidimensional extension of the Stronger Central Sets Theorem. Additionally, we develop a polynomial generalization of this result. Our ap…
Monochromatic Polynomial sumset structures on : an ultrafilter proof
Sayan Goswami
Recently, using machinery's from Ergodic theory, Z. Lian, and R. Xiao proved if is any polynomial with no constant term, then for every finite coloring of , there e…
The interplay between additive and symmetric large sets and their combinatorial applications
Arkabrata Ghosh, Sayan Goswami, Sourav Kanti Patra
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,…
On Polynomial Extensions of van der Waerden's Theorem and its Applications
Sayan Goswami
In this article, we investigate polynomial generalizations of the van der Waerden theorem with a focus on largeness properties of recurrence patterns. We prove an -stre…