Related Papers
arXiv: Combinatorics
Siblings of countable cographs
2020 •
Robert Woodrow
We show that every countable cograph has either one or infinitely many siblings. This answers, very partially, a conjecture of Thomasse. The main tools are the notion of well quasi ordering and the correspondence between cographs and some labelled ordered trees.
Siblings of an ℵ_0-categorical relational structure
2019 •
Maurice Pouzet
A sibling of a relational structure R is any structure S which can be embedded into R and, vice versa, in which R can be embedded. Let sib(R) be the number of siblings of R, these siblings being counted up to isomorphism. Thomassé conjectured that for countable relational structures made of at most countably many relations, sib(R) is either 1, countably infinite, or the size of the continuum; but even showing the special case sib(R)=1 or infinite is unsettled when R is a countable tree. This is related to Bonato-Tardif conjecture asserting that for every tree T the number of trees which are sibling of T is either one or infinite. We prove that if R is countable and ℵ_0-categorical, then indeed sib(R) is one or infinite. Furthermore, sib(R) is one if and only if R is finitely partitionable in the sense of Hodkinson and Macpherson. The key tools in our proof are the notion of monomorphic decomposition of a relational structure introduced in a paper by Pouzet and Thiéry 2013 and studie...
HAL (Le Centre pour la Communication Scientifique Directe)
Siblings of an $\aleph_0$-categorical relational structure
2021 •
Robert Woodrow
arXiv (Cornell University)
SIblings of an aleph_zero categorical relational structure
2018 •
Robert Woodrow
arXiv: Logic
Incompatible category forcing axioms
2018 •
David Asperó
Given a cardinal $\lambda$, category forcing axioms for $\lambda$-suitable classes $\Gamma$ are strong forcing axioms which completely decide the theory of the Chang model $\mathcal C_\lambda$, modulo generic extensions via forcing notions from $\Gamma$. $\mathsf{MM}^{+++}$ was the first category forcing axiom to be isolated (by the second author). In this paper we present, without proofs, a general theory of category forcings, and prove the existence of $\aleph_1$-many pairwise incompatible category forcing axioms for $\omega_1$-suitable classes.
Topology and Its Applications
Some notes concerning the hom*ogeneity of Boolean algebras and Boolean spaces
2003 •
Stefan Geschke
Siblings of an _0-categorical relational structure
2018 •
Maurice Pouzet
A sibling of a relational structure R is any structure S which can be embedded into R and, vice versa, in which R can be embedded. Let sib(R) be the number of siblings of R, these siblings being counted up to isomorphism. Thomassé conjectured that for countable relational structures made of at most countably many relations, sib(R) is either 1, countably infinite, or the size of the continuum; but even showing the special case sib(R)=1 or infinite is unsettled when R is a countable tree. This is related to Bonato-Tardif conjecture asserting that for every tree T the number of trees which are sibling of T is either one or infinite. We prove that if R is countable and _0-categorical, then indeed sib(R) is one or infinite. Furthermore, sib(R) is one if and only if R is finitely partitionable in the sense of Hodkinson and Macpherson. The key tools in our proof are the notion of monomorphic decomposition of a relational structure introduced in a paper by Pouzet and Thiéry 2013 and studied...
Journal of Mathematical Logic
Incompatible bounded category forcing axioms
David Asperó
We introduce bounded category forcing axioms for well-behaved classes [Formula: see text]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [Formula: see text] modulo forcing in [Formula: see text], for some cardinal [Formula: see text] naturally associated to [Formula: see text]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [Formula: see text] — to classes [Formula: see text] with [Formula: see text]. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms but can be forced under mild large cardinal assumptions on [Formula: see text]. We also show the existence of many classes [Formula: see text] with [Formula: see text] giving rise to pairwise incompatible theories for [Formula: see text].
L O ] 8 J an 2 02 1 INCOMPATIBLE BOUNDED CATEGORY FORCING AXIOMS
2021 •
David Asperó
We introduce bounded category forcing axioms for well-behaved classes Γ. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe Hλ+ Γ modulo forcing in Γ, for some cardinal λΓ naturally associated to Γ. These axioms naturally extend projective absoluteness for arbitrary set-forcing—in this situation λΓ = ω—to classes Γ with λΓ > ω. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms, but can be forced under mild large cardinal assumptions on V . We also show the existence of many classes Γ with λΓ = ω1, and giving rise to pairwise incompatible theories for Hω2 .
Journal of Algebra
algebras that are sharply transitive modules
2007 •
Daniel Herden
