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A cardinal trichotomy for topological equivalence classes of countable trees — arXiv:2609.27872

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.27872
License (as archived): CC0-1.0
Archived: 2026-10-08T12:23:56.103Z
SHA-256 of the archived text: e4c8eb766b5abd2a72bb0dea8a916be924fb20c3f4a8946d68f7c88a8dd8b881
Stable URL of this page: https://metraipolis.com/documento/dc_bf9bb331bec3

How to cite

A cardinal trichotomy for topological equivalence classes of countable trees — arXiv:2609.27872 — arXiv (en), https://arxiv.org/abs/2609.27872 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bf9bb331bec3 (Metraipolis archive dc_bf9bb331bec3, sha256 e4c8eb766b5abd2a72bb0dea8a916be924fb20c3f4a8946d68f7c88a8dd8b881)

BibTeX

@misc{dcbf9bb331bec3,
  title = {A cardinal trichotomy for topological equivalence classes of countable trees — arXiv:2609.27872},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.27872},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bf9bb331bec3; sha256 e4c8eb766b5abd2a72bb0dea8a916be924fb20c3f4a8946d68f7c88a8dd8b881},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.27872. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

For a tree $T$, let $[T]$ be the set of isomorphism classes of trees that are mutually topological minors of $T$. Bruno and Szeptycki proved that every locally finite tree satisfies $|[T]|\in\{1,2^{\aleph_0}\}$ and that every tree with a ray containing infinitely many vertices of degree at least $3$ has at least $2^{\aleph_0}$ topological twins. Hence equality holds in the latter case when the tree is countable. We treat the complementary countable case, without any bound on the degrees. For every such tree $T$, we prove that $|[T]|\in\{1,\aleph_0,2^{\aleph_0}\}$ and give a well-founded recursive description based on Schmidt rank. The proof uses a finite canonical subtree extracted from the subtree generated by the branching vertices and a counting theorem for countable multisets over a well-quasi-order. The same trichotomy follows for every countable tree and hence for every tree all of whose vertices have countable degree.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-24 in math.LO).
Categorie arXiv: Logic (math.LO) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.27872 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-08T12:10:31.681Z dalla pagina https://arxiv.org/catchup/math.LO/2026-09-24?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).