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Keywords
nonmeager set Minimal Walks regressive Souslin tree Filter reflection Interval topology on trees Ascending path Respecting tree Ramsey theory over partitions middle diamond stationary reflection Poset Reflecting stationary set incompactness Cardinal function Local Club Condensation. Generalized descriptive set theory club_AD Antichain Rado's conjecture Successor of Singular Cardinal Aronszajn tree Postprocessing function Weakly compact cardinal Distributive tree S-Space Almost countably chromatic Luzin set higher Baire space Rainbow sets Sierpinski's onto mapping principle specializable Souslin tree free Boolean algebra square Microscopic Approach Parameterized proxy principle Slim tree Vanishing levels Open Access Strongly Luzin set P-Ideal Dichotomy Dowker space Entangled linear order Subnormal ideal polarized partition relation Generalized Clubs Foundations b-scale Forcing Axioms Amenable C-sequence Prikry-type forcing Knaster and friends Ostaszewski square Was Ulam right? Small forcing Monotonically far Diamond-sharp Almost Souslin Subtle cardinal Chromatic number Erdos Cardinal Uniformly coherent full tree ZFC construction SNR C-sequence Nonspecial tree unbounded function Uniformization Erdos-Hajnal graphs strongly bounded groups Almost-disjoint family Shelah's Strong Hypothesis Jonsson cardinal xbox Lipschitz reduction stick Intersection model indecomposable filter Commutative cancellative semigroups Large Cardinals weak square reflection principles OCA Martin's Axiom Partition Relations Absoluteness Dushnik-Miller super-Souslin tree Rock n' Roll PFA weak diamond Whitehead Problem projective Boolean algebra weak Kurepa tree Non-saturation Singular cardinals combinatorics sap Axiom R diamond star Forcing with side conditions tensor product graph Constructible Universe free Souslin tree Singular Density Partition relations for trees Iterated forcing AIM forcing Diamond for trees approachability ideal Countryman line Greatly Mahlo Chang's conjecture Hindman's Theorem Diamond ccc L-space Cohen real perfectly normal GMA Well-behaved magma Singular cofinality 54G20 Hereditarily Lindelöf space Successor of Regular Cardinal Fat stationary set Uniformly homogeneous HOD Kurepa Hypothesis Commutative projection system Selective Ultrafilter PFA(S)[S] very good scale Club Guessing Souslin Tree Strongly compact cardinal Fodor-type reflection Subtle tree property Fast club stationary hitting Knaster Universal Sequences Analytic sets countably metacompact Ascent Path Ulam matrix Precaliber Subadditive Mandelbrot set Prevalent singular cardinals Square-Brackets Partition Relations Sakurai's Bell inequality Coherent tree Ineffable cardinal Reduced Power coloring number Cardinal Invariants O-space square principles Closed coloring Sigma-Prikry transformations Forcing Strong coloring Hedetniemi's conjecture positive partition relation
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
