论文标题
有限结构的超强型的大拉姆西学位
Big Ramsey Degrees in Ultraproducts of Finite Structures
论文作者
论文摘要
我们将结构性拉姆西理论的转移原理从有限的结构到超副产物开发。我们表明，在某些温和条件下，当一类有限的结构具有有限的小拉姆西学位时，在（广义）连续性假设下，超副作用具有有限的大拉姆西（Ramsey），用于内部着色。有限线性订单的超强示例证明了限制内部着色的必要性。在CH下，此Ultraproduct $ \ fll^*$作为脊柱，$η_1$，是订单类型$η$的无数类似物。 Devlin在\ cite {devlin}中精确地计算出$η$的有限大拉姆西学位。从\ cite {tod87}立即，$η_1$无法获得有限的大拉姆西学位。此外，我们将Devlin的着色扩展到$η_1$，以表明它在每份$η_1，$ in $η$的$η$上，以$η$的$η$中的$η$，$ $ \ fll^*$进行了$η$。这项工作还提供了额外的确认，即超副产品是研究有限和无限结构的拉姆西特性的合适环境。
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.
