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Several combinatorial results generalized from one large subset of semigroups to infinitely many

Teng Zhang

TL;DR

The paper extends Phulara's 2015 framework that transfers a central-set phenomenon from a single $C$-set to countably many into broader, multi-indexed contexts. It proves a general theorem in which a minimal idempotent $p$ in $\beta\mathbb{N}$ and a sequence of matrix systems yield coordinated $FS$-structure inside corresponding $B_{\min F}^\star$, enabling van der Waerden-type AP results, Hindman-type IP-sum results, and Hales-Jewett-type statements within central sets, with further extensions to uncountably many $C$-sets via a $\kappa^\omega$-type framework. A commutative semigroup generalization uses an idempotent $r \in J(S)$ and a map $R$ to construct $\alpha$ and $H$ so that finite-sum expressions land in $R(G_1)$ for all chains, and this framework extends to noncommutative semigroups as well. The work also identifies natural limitations—such as the failure of infinite-AP extensions in the van der Waerden sense and the infinite-alphabet Hales-Jewett analogue—mapping out the boundaries of these multi-set transfer phenomena and suggesting directions for further noncommutative and uncountable-index research.

Abstract

In 2015, Phulara established a generalization of the famous central set theorem by an original idea. Roughly speaking, this idea extends a combinatorial result from one large subset of the given semigroup to countably many. In this paper, we apply this idea to other combinatorial results to obtain corresponding generalizations, and do some further investigation. Moreover, we find that Phulara's generalization can be generalized further that can deal with uncountably many C-sets.

Several combinatorial results generalized from one large subset of semigroups to infinitely many

TL;DR

The paper extends Phulara's 2015 framework that transfers a central-set phenomenon from a single -set to countably many into broader, multi-indexed contexts. It proves a general theorem in which a minimal idempotent in and a sequence of matrix systems yield coordinated -structure inside corresponding , enabling van der Waerden-type AP results, Hindman-type IP-sum results, and Hales-Jewett-type statements within central sets, with further extensions to uncountably many -sets via a -type framework. A commutative semigroup generalization uses an idempotent and a map to construct and so that finite-sum expressions land in for all chains, and this framework extends to noncommutative semigroups as well. The work also identifies natural limitations—such as the failure of infinite-AP extensions in the van der Waerden sense and the infinite-alphabet Hales-Jewett analogue—mapping out the boundaries of these multi-set transfer phenomena and suggesting directions for further noncommutative and uncountable-index research.

Abstract

In 2015, Phulara established a generalization of the famous central set theorem by an original idea. Roughly speaking, this idea extends a combinatorial result from one large subset of the given semigroup to countably many. In this paper, we apply this idea to other combinatorial results to obtain corresponding generalizations, and do some further investigation. Moreover, we find that Phulara's generalization can be generalized further that can deal with uncountably many C-sets.

Paper Structure

This paper contains 3 sections, 12 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Several Applications
  3. Further Generalizations

Key Result

Theorem 1

Suppose $p$ is a minimal idempotent in $(\beta\mathbb{N}, +)$, $\langle B_n \rangle_{n=1}^\infty$ is a sequence in $p$, $\langle A_n \rangle_{n=1}^\infty$ is a sequence of finite image partition regular matrices with entries from $\mathbb{Q}$ and $m_n$ is the number of columns of $A_n$ for each $n \

Theorems & Definitions (23)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 13 more