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