Big Ramsey Degrees in Ultraproducts of Finite Structures

TL;DR

This paper establishes a transfer principle for structural Ramsey theory from finite structures to their ultraproducts, demonstrating conditions under which ultraproducts have finite big Ramsey degrees for internal colorings, and extending known results to uncountable order types.

Contribution

It introduces a transfer principle connecting finite and infinite structures' Ramsey properties via ultraproducts, under certain set-theoretic assumptions, and extends Devlin's results to uncountable orders.

Findings

Ultraproducts of finite structures can have finite big Ramsey degrees under CH.

Finite big Ramsey degrees for $LL^*$ are demonstrated for internal colorings.

Extension of Devlin's coloring to $LL^*$ confirms finite big Ramsey degrees in 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, $\eta_1$, an uncountable analogue of the order type of rationals $\eta$. Finite big Ramsey degrees for $\eta$ were exactly calculated by Devlin in \cite{Devlin}. It is immediate from \cite{Tod87} that $\eta_1$ fails to have finite big Ramsey degrees. Moreover, we extend Devlin's coloring to $\eta_1$ to show that it witnesses big Ramsey degrees of finite tuples in $\eta$ on every copy of $\eta$ in $\eta_1,$ and consequently in $\fLL^*$. This work gives additional confirmation that ultraproducts are a suitable environment for studying Ramsey properties of finite and infinite structures.