Big Ramsey Degrees in Ultraproducts of Finite Structures

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 ${\text{fLL}}^{\ast }$ 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 ${\text{fLL}}^{\ast }$. This work gives additional confirmation that ultraproducts are a suitable environment for studying Ramsey properties of finite and infinite structures.