Stable finitely homogeneous structures

Cherlin, G.; Lachlan, A. H.

doi:10.1090/S0002-9947-1986-0846608-8

Stable finitely homogeneous structures
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by G. Cherlin and A. H. Lachlan PDF

Trans. Amer. Math. Soc. 296 (1986), 815-850 Request permission

Abstract:

Let $L$ be a finite relational language and $\operatorname {Hom}(L,\omega )$ denote the class of countable $L$-structures which are stable and homogeneous. The main result of the paper is that there exists a natural number $c(L)$ such that for any transitive $\mathcal {M} \in \operatorname {Hom}(L;\omega )$, if $E$ is a maximal $0$-definable equivalence relation on $\mathcal {M}$, then either $|\mathcal {M}/E| < c(L)$, or $\mathcal {M}/E$ is coordinatizable. In an earlier paper the second author analyzed certain subclasses $\operatorname {Hom}(L, r)\ (r < \omega )$ of $\operatorname {Hom}(L,\omega )$ for all sufficiently small $r$. Thus the earlier analysis now applies to $\operatorname {Hom}(L,\omega )$.

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