For a structure let φ() be the number of nonisomorphic, countably infinite substructures of . The problem considered here, suggested by M. Pouzet, is that of characterizing those countable for which φ() ≤ ℵ0. In this paper we will deal exclusively with structures in a finite, binary relational language L. The characterization of those L-structures for which φ() ≤ ℵ0 (which turns out to be equivalent to ) is given in Theorem 3. It is the culmination of a three-step process. The first step, resulting in Theorem 1, shows that for a countable stable L-structure , φ() ≤ ℵ0 iff is cellular. (See Definition 0.1.) In the second step we consider linearly ordered sets = (A, ≤ ℵ0), and characterize in Theorem 2 the order types of those for which φ() ≤ ℵ0. Finally, in Theorem 3, we amalgamate Theorems 1 and 2 to get the classification of all countable L-structures for which φ() ≤ ℵ0.
We understand that Zs. Nagy independently has obtained results on countable graphs Γ for which .
Fix a finite relational language L and let be an L-structure. For a
0, a
1,…, a
n−1 ∈ A, the type of 〈a
0, a
1,…,a
n−1〉, denoted by tp(a
0,a
1,…,a
n−1), is the set of quantifier-free
L-formulas φ(x
0,x
1,…,x
n−1) for which ⊨ φ(a
0,a
1,…,a
n−1). More generally, if X ⊆ A, then the type of 〈a
0,a
1,…, a
n−1〉 over X is defined to be tp(a
0,a
1,…,a
n−1/X) = {φ(): φ() is a quantifier-free (L ∪ X)-formula such that ⊨ φ(ā)}.