A Fra\"iss\'e theory for partial orders of a fixed finite dimension

For each $n\geq 2$, we show that the class of all finite $n$-dimensional partial orders, when expanded with $n$ linear orders which realize the partial order, forms a Fraïssé class and identify its Fraïssé limit $(D_n,<,<_1,\ldots,<_n)$.

We give a finite axiomatization of this limit which specifies it uniquely up to isomorphism among countable structures, show that its class of finite substructures satisfies the Ramsey property, and conclude, by the Kechris--Pestov--Todorčević correspondence, that the automorphism group of the limit is extremely amenable.

We then identify the universal minimal flow of the automorphism group of the reduct $(D_n,<)$.

Similar results are established for the $n$-dimensional rational grid $(\mathbb{Q}^n,<)$ and its expansion by the coordinate orders.