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A weak variant of Hindman's Theorem stronger than Hilbert's Theorem

Lorenzo Carlucci

TL;DR

The paper introduces the Adjacent Hindman's Theorem ($\mathbf{AHT}$), a natural restriction of Hindman's Finite Sums Theorem requiring monochromaticity only for sums of adjacent elements under an apartness constraint. It establishes an upper bound by showing $\mathbf{AHT}_2$ is provable from $\mathbf{RT}^2_2$ and that $\mathbf{AHT}$ for arbitrary colors follows from $\forall r\mathbf{RT}^2_r$, while providing a sharp lower bound by deriving $\mathbf{IPT}^2$ from $\mathbf{AHT}$ (and extensions to $\mathbf{IPT}^2_k$). The work connects these results to stable Ramsey theory and induction schemes, showing $\mathbf{AHT}$ implies $\mathbf{I}\Sigma^0_2$ and, for arbitrary colors, implies $\mathbf{B}\Sigma^0_3$ through a chain of implications involving $\mathbf{IPT}^2$ and $\mathbf{SRT}^2_2$. These findings illuminate how structural restrictions on the sums influence the reverse mathematical strength, and suggest a broader program of Hindman-type principles guided by combinatorial trees. The paper thus situates $\mathbf{AHT}$ between known Ramsey-type principles and induction strengths, offering a framework for further exploration of Hindman-type restrictions and their hierarchies.

Abstract

Hirst investigated a slight variant of Hindman's Finite Sums Theorem -- called Hilbert's Theorem -- and proved it equivalent over $\RCA_0$ to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman's Theorem provably much weaker than Hindman's Theorem itself. We here introduce another natural variant of Hindman's Theorem -- which we name the Adjacent Hindman's Theorem -- and prove it to be provable from Ramsey's Theorem for pairs and strictly stronger than Hirst's Hilbert's Theorem. The lower bound is obtained by a direct combinatorial implication from the Adjacent Hindman's Theorem to the Increasing Polarized Ramsey's Theorem for pairs introduced by Dzhafarov and Hirst. In the Adjacent Hindman's Theorem homogeneity is required only for finite sums of adjacent elements.

A weak variant of Hindman's Theorem stronger than Hilbert's Theorem

TL;DR

The paper introduces the Adjacent Hindman's Theorem (), a natural restriction of Hindman's Finite Sums Theorem requiring monochromaticity only for sums of adjacent elements under an apartness constraint. It establishes an upper bound by showing is provable from and that for arbitrary colors follows from , while providing a sharp lower bound by deriving from (and extensions to ). The work connects these results to stable Ramsey theory and induction schemes, showing implies and, for arbitrary colors, implies through a chain of implications involving and . These findings illuminate how structural restrictions on the sums influence the reverse mathematical strength, and suggest a broader program of Hindman-type principles guided by combinatorial trees. The paper thus situates between known Ramsey-type principles and induction strengths, offering a framework for further exploration of Hindman-type restrictions and their hierarchies.

Abstract

Hirst investigated a slight variant of Hindman's Finite Sums Theorem -- called Hilbert's Theorem -- and proved it equivalent over to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman's Theorem provably much weaker than Hindman's Theorem itself. We here introduce another natural variant of Hindman's Theorem -- which we name the Adjacent Hindman's Theorem -- and prove it to be provable from Ramsey's Theorem for pairs and strictly stronger than Hirst's Hilbert's Theorem. The lower bound is obtained by a direct combinatorial implication from the Adjacent Hindman's Theorem to the Increasing Polarized Ramsey's Theorem for pairs introduced by Dzhafarov and Hirst. In the Adjacent Hindman's Theorem homogeneity is required only for finite sums of adjacent elements.

Paper Structure

This paper contains 6 sections, 7 theorems, 18 equations.

Table of Contents

  1. Introduction and Motivation
  2. The Adjacent Hindman's Theorem
  3. Upper Bound: Adjacent Hindman's Theorem follows from Ramsey for pairs
  4. Lower Bound: Adjacent Hindman's Theorem implies Increasing Polarized Ramsey's Theorem for pairs
  5. Adjacent Hindman's Theorem and $\Sigma^0_2$-induction
  6. Conclusions

Key Result

Proposition 1

Over $\mathbf{RCA}_0$, $\mathbf{RT}^2_2$ implies $\mathbf{AHT}_2$.

Theorems & Definitions (18)

  • Definition 1: Hilbert's Theorem, Hir:12:HvsH
  • Definition 2: Apartness Condition
  • Definition 3: Adjacent Hindman's Theorem
  • Proposition 1
  • proof
  • Corollary 1
  • Definition 4: Increasing Polarized Ramsey Theorem
  • Proposition 2
  • proof
  • Corollary 2
  • ...and 8 more