Fundamental theorem of linear orders

Question

Let $(\Omega,\leq)$ be a countable linear order. Suppose that for every finite $m \in \mathbb{N}$, and all subsets $S_1$ and $S_2$ of $\Omega$ of order $m$, there is an order-automorphism of $(\Omega,\leq)$ that sends $S_1$ to $S_2$.

Call the order-automorphism group $A$.

Is there a "fundamental theorem" for $(\Omega,\leq)$ in the spirit of the fundamental theorem of projective geometry, of the following form:

• if $\Omega_1$ and $\Omega_2$ are subsets of $\Omega$, order-isomorphic through an order-isomorphism $\gamma$, then there is an order-automorphism $\overline{\gamma}$ in $A$ which extends $\gamma$.

Of course, this statement is too general -- there should at least be some restrictions on $\Omega_i$ -- but I am looking for a statement "as general as possible."

Second related question: what if we remove the countability condition on $\Omega$ ?

Answer 1

As already observed, you may as well assume $(\Omega,\leq)$ is $(\mathbb{Q},\leq)$. In this case, of course one condition on $\Omega_1$ and $\Omega_2$ ensuring your desired property is finiteness.

So the interesting question is about the case when $\Omega_1$ and $\Omega_2$ are infinite. We can even just consider the special case when $\Omega_1=\Omega_2$.

Call a subset $X$ of $\mathbb{Q}$ good if every order automorphism of $X$ extends to an order automorphism of $\mathbb{Q}$. Then it follows from Theorem 2.19 of this paper by Panagiotopoulos that good subsets are extremely rare. In particular, consider the set of all subsets of $\mathbb{Q}$ as Cantor space. Then it follows from Panagiotopoulos's work that the set of good subsets of $\mathbb{Q}$ is meager (a countable union of nowhere dense sets).

In fact, it's even worse. Given a subset $X$ of $\mathbb{Q}$, we can also view the set $\text{Aut}(X)$ of order automorphisms of $X$ as a Polish space with the pointwise convergence topology. So call a set $X$ really bad if the set of automorphisms of $X$ that extend to automorphisms of $\mathbb{Q}$ is meager in $\text{Aut}(X)$. Then the result is that the set of really bad subsets of $\mathbb{Q}$ is comeager.

Using the language "generic" for properties that happen on a comeager set, the result says: "A generic automorphism of a generic subset of $\mathbb{Q}$ does not extend to an automorphism of $\mathbb{Q}$".

He also proves similar results for pairs of subsets. Moreover, the paper is not about $\mathbb{Q}$, but about general Fraisse limits.