Chain conditions in maximal models

Paul Larson; Stevo Todorčević

Displaying similar documents to “Chain conditions in maximal models”

Planting Kurepa trees and killing Jech-Кunen trees in a model by using one inaccessible cardinal

Saharon Shelah, R. Jin (1992)

Fundamenta Mathematicae

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By an $ω_{1}$ - tree we mean a tree of power $ω_{1}$ and height $ω_{1}$ . Under CH and $2^{ω_{1}} > ω_{2}$ we call an $ω_{1}$ -tree a Jech-Kunen tree if it has κ-many branches for some κ strictly between $ω_{1}$ and $2^{ω_{1}}$ . In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus $2^{ω_{1}} > ω_{2}$ that there exist Kurepa trees and there are no Jech-Kunen trees, which answers a question of [Ji2], (2) it is consistent with CH plus $2^{ω_{1}} = ω_{4}$ that there only exist Kurepa trees with $ω_{3}$ -many branches, which answers...

Pressing Down Lemma for $λ$ -trees and its applications

Hui Li, Liang-Xue Peng (2013)

Czechoslovak Mathematical Journal

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For any ordinal $λ$ of uncountable cofinality, a $λ$ -tree is a tree $T$ of height $λ$ such that $| T_{α} | < cf (λ)$ for each $α < λ$ , where $T_{α} = {x \in T : ht (x) = α}$ . In this note we get a Pressing Down Lemma for $λ$ -trees and discuss some of its applications. We show that if $η$ is an uncountable ordinal and $T$ is a Hausdorff tree of height $η$ such that $| T_{α} | \leq ω$ for each $α < η$ , then the tree $T$ is collectionwise Hausdorff if and only if for each antichain $C \subset T$ and for each limit ordinal $α \leq η$ with $cf (α) > ω$ , ${ht (c) : c \in C} \cap α$ is not stationary in $α$ . In the last part of this note, we investigate...

Rudin's Dowker space in the extension with a Suslin tree

Teruyuki Yorioka (2008)

Fundamenta Mathematicae

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We introduce a generalization of a Dowker space constructed from a Suslin tree by Mary Ellen Rudin, and the rectangle refining property for forcing notions, which modifies the one for partitions due to Paul B. Larson and Stevo Todorčević and is stronger than the countable chain condition. It is proved that Martin's Axiom for forcing notions with the rectangle refining property implies that every generalized Rudin space constructed from Aronszajn trees is non-Dowker, and that the same...

Closure for spanning trees and distant area

Jun Fujisawa, Akira Saito, Ingo Schiermeyer (2011)

Discussiones Mathematicae Graph Theory

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A k-ended tree is a tree with at most k endvertices. Broersma and Tuinstra [3] have proved that for k ≥ 2 and for a pair of nonadjacent vertices u, v in a graph G of order n with $d e g_{G} u + d e g_{G} v \geq n - 1$ , G has a spanning k-ended tree if and only if G+uv has a spanning k-ended tree. The distant area for u and v is the subgraph induced by the set of vertices that are not adjacent with u or v. We investigate the relationship between the condition on $d e g_{G} u + d e g_{G} v$ and the structure of the distant area for u and v. We prove...

Signpost systems and spanning trees of graphs

Ladislav Nebeský (2006)

Czechoslovak Mathematical Journal

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By a ternary system we mean an ordered pair $(W, R)$ , where $W$ is a finite nonempty set and $R \subseteq W \times W \times W$ . By a signpost system we mean a ternary system $(W, R)$ satisfying the following conditions for all $x, y, z \in W$ : if $(x, y, z) \in R$ , then $(y, x, x) \in R$ and $(y, x, z) \notin R$ ; if $x \neq y$ , then there exists $t \in W$ such that $(x, t, y) \in R$ . In this paper, a signpost system is used as a common description of a connected graph and a spanning tree of the graph. By a ct-pair we mean an ordered pair $(G, T)$ , where $G$ is a connected graph and $T$ is a spanning tree of $G$ . If $(G, T)$ is a ct-pair, then by...

Spanning caterpillars with bounded diameter

Ralph Faudree, Ronald Gould, Michael Jacobson, Linda Lesniak (1995)

Discussiones Mathematicae Graph Theory

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A caterpillar is a tree with the property that the vertices of degree at least 2 induce a path. We show that for every graph G of order n, either G or G̅ has a spanning caterpillar of diameter at most 2 log n. Furthermore, we show that if G is a graph of diameter 2 (diameter 3), then G contains a spanning caterpillar of diameter at most $c n^{3 / 4}$ (at most n).

Simplicity of Neretin's group of spheromorphisms

Christophe Kapoudjian (1999)

Annales de l'institut Fourier

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Denote by $𝒯_{n}$ , $n \geq 2$ , the regular tree whose vertices have valence $n + 1$ , $\partial 𝒯_{n}$ its boundary. Yu. A. Neretin has proposed a group $N_{n}$ of transformations of $\partial 𝒯_{n}$ , thought of as a combinatorial analogue of the diffeomorphism group of the circle. We show that $N_{n}$ is generated by two groups: the group $Aut (𝒯_{n})$ of tree automorphisms, and a Higman-Thompson group $G_{n}$ . We prove the simplicity of $N_{n}$ and of a family of its subgroups.

Pruning Galton–Watson trees and tree-valued Markov processes

Romain Abraham, Jean-François Delmas, Hui He (2012)

Annales de l'I.H.P. Probabilités et statistiques

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We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a tree-valued Markov process ${𝒢 (u)}$ by pruning Galton–Watson trees and an analogous process ${𝒢^{*} (u)}$ by pruning a critical or subcritical Galton–Watson tree conditioned to be infinite. Under a mild condition on offspring distributions, we show that the process ${𝒢 (u)}$ run until its ascension time has a representation in terms of ${𝒢^{*} (u)}$ . A similar result was obtained by...

Trees with α-labelings and decompositions of complete graphs into non-symmetric isomorphic spanning trees

Michael Kubesa (2005)

Discussiones Mathematicae Graph Theory

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We examine constructions of non-symmetric trees with a flexible q-labeling or an α-like labeling, which allow factorization of $K_{2 n}$ into spanning trees, arising from the trees with α-labelings.