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