Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ramsey degrees: big v. small

Dragan Mašulović

TL;DR

This work develops a categorical framework for Ramsey theory, treating big and small Ramsey degrees for objects and morphisms in categories with mild assumptions and clarifying their interrelations. It proves that small Ramsey degrees are the minima of big Ramsey degrees across suitable expansions, and establishes an additive relation for big Ramsey degrees under expansions, with equality when a self-similarity condition holds. The paper then shows that finite big Ramsey degrees are preserved under quantifier-free reducts, applying the machinery to key countable structures such as $(\mathbb{Q},<)$, the random graph, and the random tournament, including adding finitely many constants. Collectively, these results unify prior relational-structure findings and provide a principled method to transfer finiteness properties of big Ramsey degrees across expansions and reducts.

Abstract

In this paper we investigate algebraic properties of big Ramsey degrees in categories satisfying some mild conditions. As the first nontrivial consequence of the generalization we advocate in this paper we prove that small Ramsey degrees are the minima of the corresponding big ones. We also prove that big Ramsey degrees are subadditive and show that equality is enforced by an abstract property of objects we refer to as self-similarity. Finally, we apply the abstract machinery developed in the paper to show that if a countable relational structure has finite big Ramsey degrees, then so do its quantifier-free reducts. In particular, it follows that the reducts of (Q, <), the random graph, the random tournament and (Q, <, 0) all have finite big Ramsey degrees.

Ramsey degrees: big v. small

TL;DR

This work develops a categorical framework for Ramsey theory, treating big and small Ramsey degrees for objects and morphisms in categories with mild assumptions and clarifying their interrelations. It proves that small Ramsey degrees are the minima of big Ramsey degrees across suitable expansions, and establishes an additive relation for big Ramsey degrees under expansions, with equality when a self-similarity condition holds. The paper then shows that finite big Ramsey degrees are preserved under quantifier-free reducts, applying the machinery to key countable structures such as , the random graph, and the random tournament, including adding finitely many constants. Collectively, these results unify prior relational-structure findings and provide a principled method to transfer finiteness properties of big Ramsey degrees across expansions and reducts.

Abstract

In this paper we investigate algebraic properties of big Ramsey degrees in categories satisfying some mild conditions. As the first nontrivial consequence of the generalization we advocate in this paper we prove that small Ramsey degrees are the minima of the corresponding big ones. We also prove that big Ramsey degrees are subadditive and show that equality is enforced by an abstract property of objects we refer to as self-similarity. Finally, we apply the abstract machinery developed in the paper to show that if a countable relational structure has finite big Ramsey degrees, then so do its quantifier-free reducts. In particular, it follows that the reducts of (Q, <), the random graph, the random tournament and (Q, <, 0) all have finite big Ramsey degrees.

Paper Structure

This paper contains 8 sections, 28 theorems, 81 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ramsey degrees in a category
  4. Small Ramsey degrees as minima of the big ones
  5. Monotonicity of Ramsey degrees
  6. An additive property of big Ramsey degrees
  7. Reducts of relational structures
  8. Acknowledgements

Key Result

Lemma 2.1

Let $\mathbf{C}$ and $\mathbf{C}^*$ be locally small categories and let $U : \mathbf{C}^* \to \mathbf{C}$ be an expansion with unique restrictions. $(a)$ Let $A \in \mathrm{Ob}(\mathbf{C})$ and $A^*, A^*_1 \in U^{-1}(A)$. Let $f : A^*_1 \to A^*$ be a morphism. If $U(f) = \mathrm{id}_A$ then $A^* = A

Figures (2)

  • Figure 1: An $(A, B)$-diagram in $\mathbf{C}$
  • Figure 2: The setup of Theorem \ref{['bigrd.thm.1']}

Theorems & Definitions (47)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 3.1
  • Corollary 3.2
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • proof
  • Theorem 4.1
  • ...and 37 more