A family of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml1" display="inline" overflow="scroll" altimg="si1.gif"><mml:msub><mml:mrow><mml:mi>ω</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> topological types of locally finite trees

Bruno, Jorge

doi:10.1016/j.disc.2016.11.021

Cited by 4 publications

(7 citation statements)

References 3 publications

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“…First note that, as in the observation preceding the statement of the Lemma, b must map r 1,g to r 1,f since all other vertices v ∈ V (T f ) on the same level as r 1,f precede either a ray or finite bare paths of length at most M 1 < g (1). Moreover, the subdivision of e 1 dictated by g(1) above r 1,g must be mapped into the subdivision of e 1 above r 1,f dictated by f (1).…”

Section: Rooted Treesmentioning

confidence: 85%

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The Tree Alternative Conjecture Under the Topological Minor Relation

Bruno

Szeptycki

2022

Electron. J. Combin.

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The Tree Alternative Conjecture concerns the sizes of equivalence classes of trees with respect to mutual embeddable relation. Indeed, it conjectures that the number of isomorphism classes of trees mutually embeddable with a given tree $T$ is either 1 or infinite - with instances of size $\aleph_0$ and $2^{\aleph_0}$. We prove its analogue within the family of locally finite trees with respect to the topological minor relation. More precisely, we prove that for any locally finite tree $T$ the size of its equivalence class with respect to the topological minor relation can only be either $1$ or $2^{\aleph_0}$.

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Section: Rooted Treesmentioning

confidence: 85%

“…More importantly, since G and F are subdivisions of R it must be that G m 1 corresponds to the first vertex above r 1 with splitting number 2. Whence, b(r 1,g ) = r 1,f and thus f (1) = g (1). Now assume that n ∈ N and that we have the following inductive hypotheses:…”

Section: Rooted Treesmentioning

confidence: 99%

The Tree Alternative Conjecture Under the Topological Minor Relation

Bruno

Szeptycki

2022

Electron. J. Combin.

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“…Moreover, if B T has size c then it contains a perfect set and so (3)⇒(2). Finally, since each S α ≤ T 2 , we have (2)⇒ (1).…”

Section: 2mentioning

confidence: 89%

“…Since by design S n ≤ S n+1 and S n+1 ≤ S n we can deduce the same for S ω . Transfinite induction is employed in [1] to extend the spine argument to all α < ω 1 and yield the following result.…”

Section: 2mentioning

confidence: 99%

“…This was initially posed by van der Holst [8] and partially answered by Matthiesen [6] by providing an existential proof that |T /≡ | ≥ ℵ 1 , making explicit use of the well-quasi-ordering of T . The first author answered a question from [6] by giving an explicit construction of ℵ 1 topological types of locally finite trees [1]-indeed a chain of order type ω 1 was constructed. Let (X, ≤) be a well-founded partially ordered set with only finite antichains.…”

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confidence: 99%

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There are exactly $\omega _1$ topological types of locally finite trees with countably many rays

Bruno

Szeptycki

2022

Fund. Math.

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Nash-Williams showed that the collection of locally finite trees under the topological minor relation results in a well-quasi-order. As a consequence, Matthiesen proved that the number λ of topological types of locally finite tree must be uncountable. Since ℵ1 ≤ λ ≤ c, finding the exact value of λ becomes non-trivial in the absence of the Continuum Hypothesis. In this paper we address this task by showing that λ = ℵ1 for locally finite trees with countably many rays. We also partially extend this result to locally finite trees with uncountably many rays.

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The Number of Topological Types of Trees

Krill,

Pitz

2024

Combinatorica

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Two graphs are of the same topological type if they can be mutually embedded into each other topologically. We show that there are exactly $$\aleph _1$$ ℵ 1 distinct topological types of countable trees. In general, for any infinite cardinal $$\kappa $$ κ there are exactly $$\kappa ^+$$ κ + distinct topological types of trees of size $$\kappa $$ κ . This solves a problem of van der Holst from 2005.

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A family of ω1 topological types of locally finite trees

Cited by 4 publications

References 3 publications

The Tree Alternative Conjecture Under the Topological Minor Relation

The Tree Alternative Conjecture Under the Topological Minor Relation

There are exactly $\omega _1$ topological types of locally finite trees with countably many rays

The Number of Topological Types of Trees

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