# Fundamental theorem of linear orders

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 $$\Omega$$ ?

• The order satisfying your condition for $m=2$ must be dense and have no endpoints, so the only such order (up to isomorphism) is $(\mathbb Q,\leq)$. – Wojowu Sep 19 '19 at 13:32

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.