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Multidimensional Stronger Central Sets Theorem and its Polynomial Extension

Sayan Goswami, Sourav Kanti Patra

TL;DR

This paper develops a multidimensional extension of the Stronger Central Sets Theorem and its polynomial generalization within the Stone-Čech ultrafilter framework for discrete semigroups. It uses tensor products of ultrafilters and lifting lemmas to construct multidimensional polynomial configurations that lie in central-type sets, establishing a polynomial extension of the Central Sets Theorem and a polynomial Milliken–Taylor theorem. It further proves a separation principle for polynomial Milliken–Taylor systems and provides ultrafilter witnesses and corollaries. Overall, the work unifies multidimensional and polynomial Ramsey theory under the algebra of $\beta S$ and expands the toolkit for combinatorial number theory on discrete semigroups.

Abstract

We establish and fully characterize the multidimensional extension of the Stronger Central Sets Theorem. Additionally, we develop a polynomial generalization of this result. Our approach utilizes tools from the Algebra of the Stone-Čech compactification of discrete semigroups. Several applications of these results are also discussed.

Multidimensional Stronger Central Sets Theorem and its Polynomial Extension

TL;DR

This paper develops a multidimensional extension of the Stronger Central Sets Theorem and its polynomial generalization within the Stone-Čech ultrafilter framework for discrete semigroups. It uses tensor products of ultrafilters and lifting lemmas to construct multidimensional polynomial configurations that lie in central-type sets, establishing a polynomial extension of the Central Sets Theorem and a polynomial Milliken–Taylor theorem. It further proves a separation principle for polynomial Milliken–Taylor systems and provides ultrafilter witnesses and corollaries. Overall, the work unifies multidimensional and polynomial Ramsey theory under the algebra of and expands the toolkit for combinatorial number theory on discrete semigroups.

Abstract

We establish and fully characterize the multidimensional extension of the Stronger Central Sets Theorem. Additionally, we develop a polynomial generalization of this result. Our approach utilizes tools from the Algebra of the Stone-Čech compactification of discrete semigroups. Several applications of these results are also discussed.
Paper Structure (9 sections, 36 theorems, 36 equations)

This paper contains 9 sections, 36 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. A Brief Review of the Algebra of the Stone-Čech Compactifications of discrete semigroups
  3. Tensor Product of Ultrafilters
  4. The Central Sets Theorem and its Polynomial Extensions
  5. Polynomial Extension of the Central Sets Theorem
  6. Milliken-Taylor Theorem and its Twofold Extensions
  7. Polynomial Extension of the Milliken-Taylor Theorem
  8. Our Proofs
  9. Separation of the Polynomial Milliken-Taylor Systems

Key Result

Theorem 1.1

Let $r \in \mathbb{N}$, and suppose that $\mathbb{N}$ is partitioned into $r$ color classes, i.e., Then, for any finite collection of polynomials $F \subset \mathbb{P}$, there exist $a, d \in \mathbb{N}$ and some $1 \leq j \leq r$ such that

Theorems & Definitions (60)

  • Theorem 1.1: Polynomial van der Waerden Theorem
  • Definition 1.2: IP Set key-11
  • Theorem 1.3: Hindman’s Theorem
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Lemma 1.7: Lifting lemma
  • Theorem 1.8: Central Sets Theorem
  • Definition 1.9
  • Theorem 1.10: Stonger Central Sets Theorem, dhs
  • ...and 50 more