Vaught's Conjecture for Sums of Products of Rooted Trees

@misc{kurilic2026vaughts,
  title = {{Vaught's Conjecture for Sums of Products of Rooted Trees}},
  author = {Miloš S. Kurilić},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28606},
  eprint = {2609.28606},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28606},
}

TY  - UNPB
TI  - Vaught's Conjecture for Sums of Products of Rooted Trees
AU  - Miloš S. Kurilić
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28606
UR  - https://arxiv.org/abs/2609.28606
AB  - We confirm Vaught's conjecture for each partial order 𝕏=∑ _𝕀∏ _j<m_i𝕏_i,j from the closure ⟨ ⟨ 𝒞^rt⟩ _Π⟩ _Σ of the class 𝒞^rt of rooted trees under finite direct products and lexicographic sums. In addition, (a) 𝒯:=mathopTh (𝕏) is ω-categorical iff all the theories 𝒯_i,j:=mathopTh (𝕏_i,j) are ω-categorical; (b) 𝒯 satisfies VC^♯ (that is, I(𝒯)∈ {1,c}), if VC^♯ holds for all 𝒯_i,j. If 𝕏∈ ⟨ ⟨ 𝒞^rt_fa⟩ _Π⟩ _Σ, where 𝒞^rt_fa is the class of finitely axiomatizable rooted trees, then (c) 𝒯 is finitely axiomatizable and (d) 𝕐∈ mathopMod (𝒯) iff 𝕐 ≅ ∑ _𝕀∏ _j<m_i𝕐_i,j, where 𝕐_i,j∈ mathopMod (𝒯_i,j), for all indices. As a by-product we prove that (e) For each n the class 𝒞 _n of partial orders isomorphic to a direct product of n rooted trees of size >1 is first-order definable; (f) Each ω-categorical partial order from the class ⟨ ⟨ 𝒞^rt_fb⟩ _Π⟩ _Σ, where 𝒞^rt_fb is the class of finite-branching rooted trees, is finitely axiomatizable. Statement (f) is related to the results of Rosenstein (for the class of linear orders) and Schmerl (for the class of partial orders of finite width).
ER  -