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