TITLE: Classification of countable posets?
QUESTION [5 upvotes]: Is there a classification of countable posets where between each two comparable elements there is a third element between them?

REPLY [3 votes]: Here is the dummies version of why countable dense posets (meaning countable posets in which every maximal chain is densely ordered) are as complicated as countable binary relations in general, so that no reasonable classification of them is to be expected.
To any countable poset $(P,\lt)$ let us associate a countable set $A$ and an irreflexive binary relation $R\subseteq A\times A$ as follows: $A$ is the set of all minimal elements of $P$, and
$$R=\{(a,b)\in A\times A:\exists x,y\in P\ (a\lt x\lt y\wedge b\not\lt x\wedge b\lt y)\}.$$
Let's call $(A,R)$ the BRSC of $(P,\lt)$.
Proposition. Given a countable set $A$ and an irreflexive binary relation $R\subseteq A\times A$, we can construct a countable dense poset $(P,\lt)$ such that the BRSC of $(P,\lt)$ is isomorphic to $(A,R)$.
Proof. We assume that $A$ is disjoint from $A\times A\times\mathbb Q$. Let
$$P=A\cup\{(a,b,x):(a,b)\in R\wedge x\in\mathbb Q\}.$$
The ordering of $P$ has the following comparisons and no others:
$$(a,b,x)\lt(a,b,y)\text{ when }(a,b)\in R\wedge x,y\in\mathbb Q\wedge x\lt y;$$
$$a\lt(a,b,x)\text{ when }a,b\in R\wedge x\in\mathbb Q;$$
$$b\lt(a,b,x)\text{ when }(a,b)\in R\wedge x\in\mathbb Q\wedge x\gt0.$$