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The barrier Ramsey theorem

Alberto Marcone, Antonio Montalbán, Andrea Volpi

TL;DR

The paper develops a general barrier-based framework for finite Ramsey theorems where both the color-target sets and homogeneous sets must be large in barrier terms. It introduces barrier largeness via fronts, blocks, and SD-barriers, and defines Ram functions using the Veblen hierarchy to capture complexity, culminating in the main result Ram(α)^{1+γ}_{<ω} = φ_{log γ}(α · ω) for γ<α with α infinite. The work proves a Barrier Ramsey Theorem, establishes the Barrier Pigeonhole Principle, and provides matching lower and upper bounds through intricate ordinal-analytic constructions and barrier manipulations, including the critical nestedness of the fundamental-sequences system. These results extend finite Ramsey theory beyond α-largeness, offering new ordinal-analytic tools and potential implications for reverse mathematics and combinatorial set theory.

Abstract

In this paper we study a very general finite Ramsey theorem, where both the sets being colored and the homogeneous set must satisfy some largeness notion. For the homogeneous set this has already been done using the notion of $α$-largeness, where $α$ is a countable ordinal equipped with a system of fundamental sequences. To extend this approach the more appropriate notion is barrier largeness. Since the complexity of barriers can be measured by countable ordinals, we define Ramsey ordinals and, using appropriate iterations of the Veblen functions, we are able to compute them.

The barrier Ramsey theorem

TL;DR

The paper develops a general barrier-based framework for finite Ramsey theorems where both the color-target sets and homogeneous sets must be large in barrier terms. It introduces barrier largeness via fronts, blocks, and SD-barriers, and defines Ram functions using the Veblen hierarchy to capture complexity, culminating in the main result Ram(α)^{1+γ}_{<ω} = φ_{log γ}(α · ω) for γ<α with α infinite. The work proves a Barrier Ramsey Theorem, establishes the Barrier Pigeonhole Principle, and provides matching lower and upper bounds through intricate ordinal-analytic constructions and barrier manipulations, including the critical nestedness of the fundamental-sequences system. These results extend finite Ramsey theory beyond α-largeness, offering new ordinal-analytic tools and potential implications for reverse mathematics and combinatorial set theory.

Abstract

In this paper we study a very general finite Ramsey theorem, where both the sets being colored and the homogeneous set must satisfy some largeness notion. For the homogeneous set this has already been done using the notion of -largeness, where is a countable ordinal equipped with a system of fundamental sequences. To extend this approach the more appropriate notion is barrier largeness. Since the complexity of barriers can be measured by countable ordinals, we define Ramsey ordinals and, using appropriate iterations of the Veblen functions, we are able to compute them.
Paper Structure (10 sections, 50 theorems, 64 equations)

This paper contains 10 sections, 50 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Largeness Notions
  3. Systems of Fundamental Sequences
  4. Fronts, Blocks and Barriers
  5. Bridges Between Systems and Barriers
  6. Ramsey Theorem for Barriers
  7. The Barrier Pigeonhole Principle
  8. The Lower Bound
  9. The Upper Bound
  10. Nestedness of the System

Key Result

Lemma 2.5

Assume the system of fundamental sequences is nested. Let $n > 1$ and $(n_i)_{i \in \omega}$ be a sequence such that $n \le n_i$ for each $i$. Then for each ordinal $\alpha$ there exists $k \in \mathbb{N}$ such that $\alpha[n] = \alpha[n_0, \ldots, n_k]$.

Theorems & Definitions (123)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Corollary 2.7
  • Definition 2.8
  • ...and 113 more