Vaught's Conjecture for Unions of Products of Rooted Trees

Abstract: Let ${\mathcal C} ^{\rm rt}$ be the class of rooted trees and $$\langle {\mathcal C} ^{\rm rt}\rangle <em>{\dot{\cup }Π}$$ its minimal closure under isomorphism, finite direct products and finite disjoint unions. Posets from that closure are isomorphic to ${\mathbb X}= \dot{\bigcup} _{i&lt;n}\prod _{j&lt;m_i}{\mathbb X}_i<sup>j$, where ${\mathbb X}_i<sup>j$ are rooted trees. Defining ${\mathcal T}=\mathop{\rm Th} ({\mathbb X})$, $${\mathcal T} _i ^{j=\mathop{\rm} Th}({\mathbb X}_i<sup>j)$$, for $i&lt;n$ and $j&lt;m_i$, and $κ= \prod _{i&lt;n}\prod _{j&lt;m_i}I({\mathcal T} _i<sup>j)$, we have (a) Vaught's conjecture is true for ${\mathcal T}$: $I({\mathcal T})=κ$, if $κ\in { 1,ω,{\mathfrak{c}}}$, and, otherwise, $I({\mathcal T}) \in [3,ω)$; (b) ${\mathbb Y} \equiv {\mathbb X}$ iff $\;{\mathbb Y} \cong \dot{\bigcup}</em>{i&lt;n}\prod <em>{j&lt;m_i}{\mathbb Y} _i<sup>j$, where $${\mathbb Y}_i^j\equiv {\mathbb X}_i<sup>j$$, for $i&lt;n$ and $j&lt;m_i$; (c) ${\mathbb E}\preccurlyeq {\mathbb X}$ iff $\;{\mathbb E} =\dot{\bigcup}</em>{i&lt;n}\prod <em>{j&lt;m_i}{\mathbb E}_i<sup>j$, where $${\mathbb E}_i^{j\preccurlyeq} {\mathbb X}_i<sup>j$$, for $i&lt;n$ and $j&lt;m_i$; (d) ${\mathcal T}$ is atomic iff $\;{\mathcal T} _i<sup>j$, for $i&lt;n$ and $j&lt;m_i$, are atomic; then $\dot{\bigcup}</em>{i&lt;n}\prod <em>{j&lt;m_i}{\mathbb A}_i<sup>j$ is a countable atomic model of ${\mathcal T}$, where ${\mathbb A}_i<sup>j$ is a countable atomic model of ${\mathcal T} _i<sup>j$, for $i&lt;n$ and $j&lt;m_i$; (e) ${\mathcal T}$ is small iff $\;{\mathcal T} _i<sup>j$, for $i&lt;n$ and $j&lt;m_i$, are small; then $\dot{\bigcup}</em>{i&lt;n}\prod _{j&lt;m_i}{\mathbb S}_i<sup>j$ is a countably saturated model of ${\mathcal T}$, where ${\mathbb S}_i<sup>j$ is a countably saturated model of ${\mathcal T}_i<sup>j$, for $i&lt;n$ and $j&lt;m_i$.

• The paper proves Vaught’s conjecture for every finite disjoint union of finite products of rooted trees, with the countable-model spectrum determined by the factor spectra and excluding exactly three models.
• The paper introduces perfect pairs, an axiomatic framework showing when factorization, elementary equivalence, embeddings, atomicity, saturation, and model counts transfer from component structures to products or unions.
• Definability of product strata recovers the number of nontrivial factors and forces elementary maps to decompose componentwise, settling cases such as the product of two infinitely branching rooted trees while leaving sharper finite-spectrum questions open.

Overview and context

The paper under review addresses Vaught's conjecture — the statement that a complete countable first-order theory $T$ has either at most $\omega$ or exactly $2^{\aleph_0}$ non-isomorphic countable models — for a broad class of partial orders built from rooted trees. The class in question is $\langle\mathcal C^{\rm rt}\rangle_{\dot\cup\Pi}$, the minimal class containing all rooted trees and closed under isomorphism, finite direct products, and finite disjoint unions. Every member has the normal form

$$\mathbb X = \dot\bigcup_{i<n}\prod_{j<m_i}\mathbb X_i^{\,j},$$

with each factor $\mathbb X_i^{\,j}$ a rooted tree. The paper's central result (Theorem 6.1) establishes that Vaught's conjecture holds for the theory of every such poset: writing $I(\mathcal T)$ for the number of countable models and $$\kappa=\prod_{i<n}\prod_{j<m_i}I(\mathcal T_i^{\,j})$$, one has $I(\mathcal T)=\kappa$ whenever $\kappa\in\{1,\omega,\mathfrak c\}$, and $\omega$0 otherwise. This extends earlier work of Kurilić on "sharp" versions of the conjecture for linear orders and for products of linear orders [(Kurilić, 6 Jan 2026)-related prior work], and it removes a significant restriction of that earlier program: previously the key hypothesis was that deleting the root leaves finitely many components, which excluded natural examples such as $\omega$1.

The paper also proves structural characterizations that are of independent interest: elementary equivalence and elementary substructure relations between such posets reduce componentwise to the factors, atomicity and smallness of the theory are equivalent to atomicity and smallness of the factor theories, and the countable atomic and countably saturated models have the expected product/union form.

The abstract framework of perfect pairs

A substantial methodological contribution is an axiomatic abstraction. For an operation $\omega$2 assigning to each finite tuple of $\omega$3-structures a new $\omega$4-structure satisfying the size condition $\omega$5, the pair $\omega$6 is called perfect if three biconditionals hold:

• Unique factorization: $\omega$7 iff $\omega$8 and the factors agree up to permutation;
• Elementary equivalence: $\omega$9 iff $2^{\aleph_0}$0 with $2^{\aleph_0}$1;
• Elementary substructures: $2^{\aleph_0}$2 iff $2^{\aleph_0}$3 with $2^{\aleph_0}$4.

With saturation preservation added ($2^{\aleph_0}$5), Theorem 3.1 derives the full package: $2^{\aleph_0}$6 when the product is infinite or 1; Vaught's conjecture transfers from the factors to the composite; atomicity, smallness, categoricity transfer; and the prime and saturated models are composites of those of the factors. A useful reduction (Proposition 3.2) shows that the elementary-equivalence clause follows from the other two plus preservation of equivalence itself. The author notes explicitly that the framework does not require $2^{\aleph_0}$7 to be closed under $2^{\aleph_0}$8, which matters because the target classes here are not $2^{\aleph_0}$9-closed in a strict ZFC sense (see below).

Two motivating counterexamples show these properties genuinely fail in general: Boolean algebras violate unique factorization ($\langle\mathcal C^{\rm rt}\rangle_{\dot\cup\Pi}$0), and both $\langle\mathcal C^{\rm rt}\rangle_{\dot\cup\Pi}$1 and the countable atomless Boolean algebra elementarily embed into their squares without being products of elementary substructures.

Products of rooted trees

The technical core concerns $\langle\mathcal C^{\rm rt}\rangle_{\dot\cup\Pi}$2, rooted trees of size greater than 1. The pair $\langle\mathcal C^{\rm rt}\rangle_{\dot\cup\Pi}$3 is shown to be strongly perfect. The proof rests on a definable stratification of a product $\langle\mathcal C^{\rm rt}\rangle_{\dot\cup\Pi}$4: the sets $\langle\mathcal C^{\rm rt}\rangle_{\dot\cup\Pi}$5 of tuples exceeding exactly $\langle\mathcal C^{\rm rt}\rangle_{\dot\cup\Pi}$6 roots are definable by explicit formulas $\langle\mathcal C^{\rm rt}\rangle_{\dot\cup\Pi}$7, and within $\langle\mathcal C^{\rm rt}\rangle_{\dot\cup\Pi}$8 the blocks $\langle\mathcal C^{\rm rt}\rangle_{\dot\cup\Pi}$9 (where only coordinate $$\mathbb X = \dot\bigcup_{i<n}\prod_{j<m_i}\mathbb X_i^{\,j},$$0 exceeds its root) are separated by a definable equivalence relation $$\mathbb X = \dot\bigcup_{i<n}\prod_{j<m_i}\mathbb X_i^{\,j},$$1. From this, Theorem 4.1 obtains:

• First-order definability of arity: sentences $$\mathbb X = \dot\bigcup_{i<n}\prod_{j<m_i}\mathbb X_i^{\,j},$$2 force any structure elementarily equivalent to $$\mathbb X = \dot\bigcup_{i<n}\prod_{j<m_i}\mathbb X_i^{\,j},$$3 to satisfy $$\mathbb X = \dot\bigcup_{i<n}\prod_{j<m_i}\mathbb X_i^{\,j},$$4, so the number of factors is invariant under elementary equivalence.
• Unique factorization up to permutation, with the strong refinement that every isomorphism $$\mathbb X = \dot\bigcup_{i<n}\prod_{j<m_i}\mathbb X_i^{\,j},$$5 decomposes as $$\mathbb X = \dot\bigcup_{i<n}\prod_{j<m_i}\mathbb X_i^{\,j},$$6 for coordinatewise isomorphisms $$\mathbb X = \dot\bigcup_{i<n}\prod_{j<m_i}\mathbb X_i^{\,j},$$7.
• Componentwise elementary substructures (Theorem 4.2): if $$\mathbb X = \dot\bigcup_{i<n}\prod_{j<m_i}\mathbb X_i^{\,j},$$8, then $$\mathbb X = \dot\bigcup_{i<n}\prod_{j<m_i}\mathbb X_i^{\,j},$$9 with $\mathbb X_i^{\,j}$0. The argument uses the sentence $\mathbb X_i^{\,j}$1 asserting existence of suprema of $\mathbb X_i^{\,j}$2 pairwise incompatible elements of $\mathbb X_i^{\,j}$3, together with a uniqueness lemma showing that an elementary substructure is determined by its trace on $\mathbb X_i^{\,j}$4.

Consequently $\mathbb X_i^{\,j}$5 holds, and Theorem 4.3 extends everything to products allowing singleton factors: singleton coordinates contribute trivial theories with $\mathbb X_i^{\,j}$6, atomicity, and smallness, so they can be adjoined freely without affecting the conclusions. In particular, Vaught's conjecture holds for the theory of every finite product of rooted trees — resolving the motivating question about $\mathbb X_i^{\,j}$7 affirmatively.

Disjoint unions of structures of finite diameter

Section 5 treats the operation $\mathbb X_i^{\,j}$8 on the class $\mathbb X_i^{\,j}$9 of binary-relational structures of finite diameter. Here connectivity is first-order definable via the path formulas $I(\mathcal T)$0, and connected components coincide with maximal connected parts. Theorem 5.1 proves that $I(\mathcal T)$1 is strongly perfect: the number of components is determined by the theory, and both $I(\mathcal T)$2 and $I(\mathcal T)$3 decompose componentwise. Hence all conclusions of the abstract framework apply to disjoint unions of finite-diameter structures. Note that each product of rooted trees has diameter at most 2 (it has a least element), so this section applies directly to the summands appearing in the normal form.

The main theorem for the full closure

Combining the two operations, Theorem 6.1 gives the complete analysis for $I(\mathcal T)$4. Elementary equivalence and elementary embedding are characterized componentwise through the tree factors; the number of countable models satisfies $I(\mathcal T)$5 for $I(\mathcal T)$6 and lies in $I(\mathcal T)$7 otherwise (the exclusion of 3 following from Vaught's theorem); atomicity and smallness transfer from the factors; and the countable atomic model is $I(\mathcal T)$8 while the countably saturated model is $I(\mathcal T)$9. Since Steel proved Vaught's conjecture for trees, the conjecture holds throughout the closure.

One subtlety deserves emphasis: strictly speaking, $$\kappa=\prod_{i<n}\prod_{j<m_i}I(\mathcal T_i^{\,j})$$0 is not closed under isomorphism in ZFC, since transporting a product along an arbitrary bijection of domains yields an isomorphic copy whose domain is not a set of tuples. The author resolves this by observing that the $$\kappa=\prod_{i<n}\prod_{j<m_i}I(\mathcal T_i^{\,j})$$1-closure coincides with the $$\kappa=\prod_{i<n}\prod_{j<m_i}I(\mathcal T_i^{\,j})$$2-closure,

$$\kappa=\prod_{i<n}\prod_{j<m_i}I(\mathcal T_i^{\,j})$$3

and all results extend to the latter class. This is a genuine set-theoretic artifact rather than a substantive gap, but it does mean the statements must be read modulo this identification.

Limitations and open questions

Several restrictions bound the scope of the results. First, unique factorization (clause (a) of the abstract framework) fails for products allowing singleton factors, which is why the theory passes through $$\kappa=\prod_{i<n}\prod_{j<m_i}I(\mathcal T_i^{\,j})$$4; the resulting characterization of models of $$\kappa=\prod_{i<n}\prod_{j<m_i}I(\mathcal T_i^{\,j})$$5 is correspondingly weaker for such products. Second, the disjoint-union result requires finite diameter; the paper does not treat disjoint unions of arbitrary rooted trees, where connectivity is not uniformly definable. Third, the framework covers only finite products and finite unions; infinite analogues are not addressed. Finally, the paper confirms Vaught's conjecture but not VC$$\kappa=\prod_{i<n}\prod_{j<m_i}I(\mathcal T_i^{\,j})$$6: the value $$\kappa=\prod_{i<n}\prod_{j<m_i}I(\mathcal T_i^{\,j})$$7 is left undetermined beyond the bounds given, and determining exactly which finite values occur as $$\kappa=\prod_{i<n}\prod_{j<m_i}I(\mathcal T_i^{\,j})$$8 for $$\kappa=\prod_{i<n}\prod_{j<m_i}I(\mathcal T_i^{\,j})$$9 remains open, as does the extension to closures involving lexicographic sums or other constructions beyond products and disjoint unions.

Conclusion

The paper establishes Vaught's conjecture for the theories of all finite disjoint unions of finite products of rooted trees, via a reusable axiomatic framework ("perfect pairs") that isolates the conditions under which model-theoretic data — spectrum cardinalities, atomicity, smallness, prime and saturated models — transfer from factors to composites. The definability analysis of products of rooted trees, yielding first-order recovery of the number of factors and componentwise decomposition of elementary maps, is the principal technical achievement, and it substantially enlarges the class of partial orders for which Vaught's conjecture is settled.