We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the Ne\v{s}et\v{r}il-R\"odl Theorem and the second and third authors' Ramsey theorem for finite models (that is, structures with both relations and functions). We also find subclasses with the ordering property. For languages with relational symbols and unary functions we also show the extension property for partial automorphisms (EPPA) of free amalgamation classes. These general results solve several conjectures and provide an easy Ramseyness test for many classes of structures.

中文翻译：

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有限结构类的拉姆齐性质和扩展部分自同构

我们表明，当通过顶点的自由线性排序丰富时，具有关系和（对称）偏函数的有限结构的每个自由合并类都是 Ramsey 类。这是 Ne\v{s}et\v{r}il-R\"odl 定理和第二和第三作者的有限模型（即具有关系和函数的结构）的拉姆齐定理的共同强化。我们还找到了具有排序属性的子类。对于具有关系符号和一元函数的语言，我们还展示了自由合并类的部分自同构 (EPPA) 的扩展属性。这些一般结果解决了几个猜想，并为许多类提供了一个简单的 Ramseyness 测试结构。