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A new perspective on semi-retractions and the Ramsey property

Dana Bartošová, Lynn Scow

Abstract

We investigate the notion of a semi-retraction between two first order structures (in typically different signatures) that was introduced by the second author as a link between the Ramsey property and generalized indiscernible sequences. We further these connections between combinatorics and model theory, and look at semi-retractions through a new lens establishing transfers of the Ramsey property and finite Ramsey degrees under quite general conditions that are optimal as demonstrated by counterexamples. Finally, we compare semi-retractions to the category theoretic notion of a pre-adjunction.

A new perspective on semi-retractions and the Ramsey property

Abstract

We investigate the notion of a semi-retraction between two first order structures (in typically different signatures) that was introduced by the second author as a link between the Ramsey property and generalized indiscernible sequences. We further these connections between combinatorics and model theory, and look at semi-retractions through a new lens establishing transfers of the Ramsey property and finite Ramsey degrees under quite general conditions that are optimal as demonstrated by counterexamples. Finally, we compare semi-retractions to the category theoretic notion of a pre-adjunction.
Paper Structure (21 sections, 33 theorems, 56 equations)

This paper contains 21 sections, 33 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structural Ramsey theory and topological dynamics
  4. Semi-retractions
  5. The modeling property
  6. Semi-retractions, Reducts and Examples
  7. (Counter)example
  8. Semi-retracts of the countable atomless Boolean algebra
  9. Formula-free approach to transferring the RP
  10. Semi-retractions and categorical notions
  11. Pre-adjunctions
  12. Pre-adjunctions from semi-retractions
  13. Semi-retractions from pre-adjunctions
  14. Concluding Remarks
  15. Interpretations and the Ramsey property
  16. ...and 6 more sections

Key Result

Proposition 2.11

Fix a signature $L$ and a locally finite $L$-structure $M$. Let $\mathcal{K}:=\textrm{age}(\mathcal{M})$. Then, for any finite substructures $A, B \subseteq \mathcal{M}$, $(A,B)$ is a Ramsey duo for $\mathcal{K}$ if and only if $(A,B)$ is a Ramsey duo for $\mathcal{M}$.

Theorems & Definitions (108)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Example 2.7
  • Definition 2.8
  • Definition 2.10
  • Proposition 2.11
  • ...and 98 more