2023/10/15 by Miloš S. Kurilić, Kurilić, Miloš S., Stevo Todorčević +1
Computer Science · Mathematics · #03C15 #03C50 #03E40 #06A06 #Advanced Graph Theory Research #Advanced Topology and Set Theory #Artificial Intelligence in Games #FOS: Mathematics #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2310.09860
openalex publication_date 2023/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The poset of copies of a relational structure \mathbb X is the partial order \mathbb P (\mathbb X ) := ⟨ \ Y ⊂ X: \mathbb Y ≅ \mathbb X\, ⊂ ⟩ and each similarity of such posets (e.g. isomorphism, forcing equivalence) determines a classification of structures. We consider the countable ultrahomogeneous tournaments: \mathbb Q (the rational line), \mathbb S (2) (the circular tournament), and \mathbb T ^∞ (the random tournament); as well as the ultrahomogeneous digraphs \mathbb S (3), \mathbb Q [\mathbb In], \mathbb S (2)[\mathbb In] and \mathbb T ^∞ [\mathbb In] from Cherlin's list. If \mathbb G Rado (resp. \mathbb Q n) denotes the countable homogeneous universal graph (resp. n-labeled linear order), it turns out that \mathbb P (\mathbb T ^∞)≅ \mathbb P (\mathbb GRado) and that \mathbb P (\mathbb Q n) densely embeds in \mathbb P (\mathbb S (n)), for n∈\ 2,3\. Consequently, \mathbb B \mathbb X ≅ ro (\mathbb S ∗ π), where \mathbb S is the Sacks forcing and 1\mathbb S \Vdash "π is a separative, atomless and σ-closed forcing", whenever \mathbb X is a countable structure equimorphic with \mathbb Q, \mathbb Q n, \mathbb S (2), \mathbb S (3), \mathbb Q [\mathbb In] or \mathbb S (2)[\mathbb In]. Also, \mathbb B \mathbb X ≅ ro (\mathbb S ∗ π), where 1\mathbb S \Vdash "π is an ω-distributive forcing", whenever \mathbb X is a countable graph embedding \mathbb G Rado, or a countable tournament embedding \mathbb T ^∞, or \mathbb X =\mathbb T ^∞ [\mathbb In].