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Kurepa Hypothesis Diamond Respecting tree Well-behaved magma Reflecting stationary set Nonspecial tree Antichain Hedetniemi's conjecture diamond star Open Access Chromatic number Parameterized proxy principle Generalized descriptive set theory Almost-disjoint family Commutative projection system Prevalent singular cardinals Interval topology on trees AIM forcing Axiom R tensor product graph weak Kurepa tree Fat stationary set Vanishing levels Singular cardinals combinatorics positive partition relation Successor of Singular Cardinal PFA(S)[S] Subtle cardinal Hindman's Theorem Diamond for trees Dushnik-Miller Amenable C-sequence strongly bounded groups Knaster and friends Luzin set Monotonically far Whitehead Problem Closed coloring Lipschitz reduction transformations Ascending path Erdos Cardinal regressive Souslin tree Generalized Clubs Prikry-type forcing Diamond-sharp specializable Souslin tree stationary hitting Ostaszewski square Forcing unbounded function Commutative cancellative semigroups ccc middle diamond Small forcing Coherent tree Rado's conjecture Fast club P-Ideal Dichotomy Successor of Regular Cardinal Was Ulam right? Subnormal ideal Strongly compact cardinal Slim tree Aronszajn tree Greatly Mahlo coloring number Knaster Minimal Walks Almost Souslin Ineffable cardinal Poset Square-Brackets Partition Relations super-Souslin tree Precaliber stick Fodor-type reflection full tree Strongly Luzin set approachability ideal S-Space higher Baire space Strong coloring Sakurai's Bell inequality reflection principles GMA Ulam matrix Local Club Condensation. Rock n' Roll C-sequence 54G20 Absoluteness Partition Relations Universal Sequences Shelah's Strong Hypothesis Uniformization indecomposable filter projective Boolean algebra Sigma-Prikry ZFC construction Non-saturation Dowker space PFA weak diamond O-space Chang's conjecture free Souslin tree Uniformly homogeneous club_AD Analytic sets stationary reflection Constructible Universe Cardinal Invariants polarized partition relation Microscopic Approach Reduced Power Ascent Path Countryman line free Boolean algebra weak square Large Cardinals Subtle tree property HOD OCA nonmeager set Forcing Axioms Foundations Ramsey theory over partitions Erdos-Hajnal graphs L-space Forcing with side conditions Sierpinski's onto mapping principle Souslin Tree SNR Mandelbrot set Selective Ultrafilter incompactness very good scale Martin's Axiom Rainbow sets square Uniformly coherent sap Weakly compact cardinal square principles Iterated forcing Intersection model Distributive tree Singular Density Filter reflection xbox countably metacompact Hereditarily Lindelöf space Cohen real Subadditive Postprocessing function Jonsson cardinal Almost countably chromatic Club Guessing Singular cofinality Cardinal function perfectly normal Entangled linear order b-scale
Tag Archives: 03E35
Weak square and stationary reflection
Joint work with Gunter Fuchs. Abstract. It is well-known that the square principle $\square_\lambda$ entails the existence of a non-reflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not. Here we show that if $\mu^{cf(\lambda)}<\lambda$ for all $\mu<\lambda$, … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, stationary reflection, weak square
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A forcing axiom deciding the generalized Souslin Hypothesis
Joint work with Chris Lambie-Hanson. Abstract. We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $\lambda$, … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E57, Diamond, Forcing Axioms, Souslin Tree, square, super-Souslin tree
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Reflection on the coloring and chromatic numbers
Joint work with Chris Lambie-Hanson. Abstract. We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) … Continue reading
Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Chang's conjecture, Chromatic number, coloring number, Fodor-type reflection, incompactness, Iterated forcing, Parameterized proxy principle, Postprocessing function, Rado's conjecture, square, stationary reflection
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Strong failures of higher analogs of Hindman’s Theorem
Joint work with David J. Fernández Bretón. Abstract. We show that various analogs of Hindman’s Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that … Continue reading
Posted in Groups, Partition Relations, Publications
Tagged 03E02, 03E35, 03E75, 05A17, 05D10, 11P99, 20M14, Chang's conjecture, Commutative cancellative semigroups, Entangled linear order, Erdos Cardinal, Hindman's Theorem, Jonsson cardinal, Kurepa Hypothesis, Square-Brackets Partition Relations, Weakly compact cardinal, ZFC construction
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Higher Souslin trees and the GCH, revisited
Abstract. It is proved that for every uncountable cardinal $\lambda$, GCH+$\square(\lambda^+)$ entails the existence of a $\text{cf}(\lambda)$-complete $\lambda^+$-Souslin tree. In particular, if GCH holds and there are no $\aleph_2$-Souslin trees, then $\aleph_2$ is weakly compact in Godel’s constructible universe, improving … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, Open Access, regressive Souslin tree, Souslin Tree, square, Weakly compact cardinal, xbox
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A microscopic approach to Souslin-tree constructions. Part I
Joint work with Ari Meir Brodsky. Abstract. We propose a parameterized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged 03E05, 03E35, 03E65, 05C05, Coherent tree, Diamond, Microscopic Approach, Parameterized proxy principle, Slim tree, Souslin Tree, square, xbox
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Reduced powers of Souslin trees
Joint work with Ari Meir Brodsky. Abstract. We study the relationship between a $\kappa$-Souslin tree $T$ and its reduced powers $T^\theta/\mathcal U$. Previous works addressed this problem from the viewpoint of a single power $\theta$, whereas here, tools are developed … Continue reading
Same Graph, Different Universe
Abstract. May the same graph admit two different chromatic numbers in two different universes? how about infinitely many different values? and can this be achieved without changing the cardinals structure? In this paper, it is proved that in Godel’s constructible … Continue reading
Posted in Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, approachability ideal, Chromatic number, Constructible Universe, Forcing, Ostaszewski square
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Chromatic numbers of graphs – large gaps
Abstract. We say that a graph $G$ is $(\aleph_0,\kappa)$-chromatic if $\text{Chr}(G)=\kappa$, while $\text{Chr}(G’)\le\aleph_0$ for any subgraph $G’$ of $G$ of size $<|G|$. The main result of this paper reads as follows. If $\square_\lambda+\text{CH}_\lambda$ holds for a given uncountable cardinal $\lambda$, … Continue reading
Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Almost countably chromatic, Chromatic number, incompactness, Ostaszewski square
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Jensen’s diamond principle and its relatives
This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127). Abstract: We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club … Continue reading
