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Big Ramsey Degrees in Ultraproducts of Finite Structures

Dana Bartošová, Mirna Džamonja, Rehana Patel, Lynn Scow

TL;DR

This work develops a transfer principle from finite structural Ramsey properties to ultraproducts, showing that under (Generalized) Continuum Hypothesis, finite small Ramsey degrees in the age can yield finite internal big Ramsey degrees for internal colorings in the ultraproduct. It introduces internal colorings as a robust tool for transferring Ramsey properties and proves an ω-case main theorem, with generalization to larger cardinals and a corollary under CH/GCH. A detailed, concrete analysis of linear orders under CH, via the spine $\eta_1$ and Devlin-type colorings on $\mathfrak L^*$, demonstrates both the reach and the limits of the transfer principle, including counterexamples for external colorings. The results advocate ultraproducts as a natural framework for studying Ramsey properties across finite and infinite structures and connect to deep classical partition results for uncountable orders.

Abstract

We develop a transfer principle of structural Ramsey theory from finite structures to ultraproducts. We show that under certain mild conditions, when a class of finite structures has finite small Ramsey degrees, under the (Generalized) Continuum Hypothesis the ultraproduct has finite big Ramsey degrees for internal colorings. The necessity of restricting to internal colorings is demonstrated by the example of the ultraproduct of finite linear orders. Under CH, this ultraproduct $\fLL^*$ has, as a spine, $η_1$, an uncountable analogue of the order type of rationals $η$. Finite big Ramsey degrees for $η$ were exactly calculated by Devlin in \cite{Devlin}. It is immediate from \cite{Tod87} that $η_1$ fails to have finite big Ramsey degrees. Moreover, we extend Devlin's coloring to $η_1$ to show that it witnesses big Ramsey degrees of finite tuples in $η$ on every copy of $η$ in $η_1,$ and consequently in $\fLL^*$. This work gives additional confirmation that ultraproducts are a suitable environment for studying Ramsey properties of finite and infinite structures.

Big Ramsey Degrees in Ultraproducts of Finite Structures

TL;DR

This work develops a transfer principle from finite structural Ramsey properties to ultraproducts, showing that under (Generalized) Continuum Hypothesis, finite small Ramsey degrees in the age can yield finite internal big Ramsey degrees for internal colorings in the ultraproduct. It introduces internal colorings as a robust tool for transferring Ramsey properties and proves an ω-case main theorem, with generalization to larger cardinals and a corollary under CH/GCH. A detailed, concrete analysis of linear orders under CH, via the spine and Devlin-type colorings on , demonstrates both the reach and the limits of the transfer principle, including counterexamples for external colorings. The results advocate ultraproducts as a natural framework for studying Ramsey properties across finite and infinite structures and connect to deep classical partition results for uncountable orders.

Abstract

We develop a transfer principle of structural Ramsey theory from finite structures to ultraproducts. We show that under certain mild conditions, when a class of finite structures has finite small Ramsey degrees, under the (Generalized) Continuum Hypothesis the ultraproduct has finite big Ramsey degrees for internal colorings. The necessity of restricting to internal colorings is demonstrated by the example of the ultraproduct of finite linear orders. Under CH, this ultraproduct has, as a spine, , an uncountable analogue of the order type of rationals . Finite big Ramsey degrees for were exactly calculated by Devlin in \cite{Devlin}. It is immediate from \cite{Tod87} that fails to have finite big Ramsey degrees. Moreover, we extend Devlin's coloring to to show that it witnesses big Ramsey degrees of finite tuples in on every copy of in and consequently in . This work gives additional confirmation that ultraproducts are a suitable environment for studying Ramsey properties of finite and infinite structures.
Paper Structure (14 sections, 24 theorems, 53 equations)

This paper contains 14 sections, 24 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model-theoretic basics
  4. Ultrafilters and ultraproducts
  5. Partition properties and internal colorings
  6. Small and big Ramsey degrees
  7. Internal colorings
  8. Big Ramsey degrees for internal colorings: $\omega$ case
  9. Big Ramsey degrees for internal colorings: general case
  10. The case of linear orders under CH
  11. Some history around $\eta_1$ and partition properties
  12. Witnessing Devlin types in all copies of $\eta$ in $\mathfrak{L}^*$
  13. Open problems and further directions
  14. Appendix: Saturation for types with many variables

Key Result

Theorem 2.13

Let $\kappa$ be an infinite cardinal and let ${\mathfrak A}$, ${\mathfrak B}$ be ${\mathcal{L}}$-structures. If ${\mathfrak A} \equiv {\mathfrak B}$, $|A|=|B|=\kappa$, and both ${\mathfrak A}$ and ${\mathfrak B}$ are $\kappa$-saturated, then ${\mathfrak A} \cong {\mathfrak B}$.

Theorems & Definitions (68)

  • Definition 1.2
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • Remark 2.12
  • ...and 58 more