Question on countably homogeneous structures

Question

A homogeneous structure is a countable first order structure $M$ over a relational language such that any isomorphism between finite substructures of $M$ can be extended to an automorphism of $M$.

Lachlan proved that if $M$ is any stable countably homogeneous structure over a finite relational language $\mathcal{L}$, then $M$ is a union of a chain $\{M_n : n ∈ \Bbb{N}\}$ of finite homogeneous $\mathcal{L}$-structures, and each sentence $σ ∈ Th(M)$ holds in all but finitely many of $M_n$.

I think that Lachlan proved this theorem based upon graph theory because he mainly works on model based graph theory. I wonder which of Lachlan's papers (or papers/books by others) contains the proof of this theorem.

Answer 1

Let's first observe that any stable countable homogeneous structure in a finite relational language is $\aleph_0$-categorical and $\aleph_0$-stable. This is explained in A survey of homogeneous structures by Macpherson: $\aleph_0$-categoricity is Corollary 3.1.3 on p. 17, and $\aleph_0$-stability is in the paragraph just before Example 3.3.2 on p. 22.

Then the result in your question follows directly from Corollary 7.4 in $\aleph_0$-categorical, $\aleph_0$-stable structures, by Cherlin, Harrington, and Lachlan.

I'm sure that the result also follows from the work in the pair of papers On countable structures which are homogeneous for a finite relational language by Lachlan and Stable finitely homogeneous structures by Cherlin and Lachlan. But after taking a quick look at those papers, I wasn't able to find a clear statement of the result in the form you quoted it.

By the way, it's completely wrong to think that Lachlan is "mainly a graph theorist". He is a first-rate logician, and while he did a lot of work at the intersection of model theory and combinatorics, he has broad interests in pure model theory and computability theory. The work on stable homogeneous structures does not use graph theory (though the results do apply to certain theories of graphs). Instead, this work uses a lot of group theory, including the classification of finite simple groups.