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