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Minimal Walks HOD incompactness Prikry-type forcing Singular Density Singular cardinals combinatorics S-Space Martin's Axiom Cohen real club_AD free Boolean algebra Fast club Filter reflection Diamond square coloring number Strong coloring Hedetniemi's conjecture tensor product graph Ostaszewski square Erdos-Hajnal graphs Axiom R Forcing Square-Brackets Partition Relations regressive Souslin tree Selective Ultrafilter Uniformly coherent projective Boolean algebra full tree very good scale weak diamond Partition Relations Antichain approachability ideal b-scale Coherent tree Uniformization Ascent Path Large Cardinals Forcing Axioms Precaliber sap Strongly Luzin set Rock n' Roll Non-saturation Microscopic Approach Fodor-type reflection Chromatic number 54G20 positive partition relation Local Club Condensation. Amenable C-sequence SNR L-space Rainbow sets Ulam matrix polarized partition relation Prevalent singular cardinals indecomposable ultrafilter Was Ulam right Closed coloring nonmeager set Slim tree Subadditive ZFC construction Almost countably chromatic transformations P-Ideal Dichotomy higher Baire space Subtle tree property C-sequence Greatly Mahlo Whitehead Problem Open Access Postprocessing function Distributive tree Knaster Lipschitz reduction PFA(S)[S] square principles strongly bounded groups Absoluteness Successor of Regular Cardinal Weakly compact cardinal Diamond-sharp Analytic sets Luzin set PFA Hereditarily Lindelöf space Kurepa Hypothesis middle diamond Uniformly homogeneous unbounded function Subtle cardinal Well-behaved magma Commutative cancellative semigroups Universal Sequences Generalized descriptive set theory Mandelbrot set Reflecting stationary set Knaster and friends GMA Dushnik-Miller Poset Cardinal Invariants Rado's conjecture Souslin Tree Almost Souslin Jonsson cardinal reflection principles super-Souslin tree Ramsey theory over partitions O-space diamond star Sigma-Prikry OCA Iterated forcing Singular cofinality Successor of Singular Cardinal Erdos Cardinal Hindman's Theorem Parameterized proxy principle ccc Club Guessing Small forcing Nonspecial tree Ineffable cardinal free Souslin tree Vanishing levels Aronszajn tree weak square Chang's conjecture specializable Souslin tree Reduced Power AIM forcing Sakurai's Bell inequality Shelah's Strong Hypothesis weak Kurepa tree Constructible Universe Foundations Sierpinski's onto mapping principle stationary reflection Diamond for trees Cardinal function stick stationary hitting xbox Subnormal ideal Generalized Clubs Almost-disjoint family Dowker space Fat stationary set countably metacompact
Tag Archives: 03E35
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, 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
16 Comments
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
5 Comments
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
10 Comments
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
6 Comments
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
A cofinality-preserving small forcing may introduce a special Aronszajn tree
Extended Abstract: Shelah proved that Cohen forcing introduces a Souslin tree; Jensen proved that a c.c.c. forcing may consistently add a Kurepa tree; Todorcevic proved that a Knaster poset may already force the Kurepa hypothesis; Irrgang introduced a c.c.c. notion … Continue reading
Posted in Publications, Squares and Diamonds
Tagged 03E04, 03E05, 03E35, Aronszajn tree, Small forcing, Successor of Singular Cardinal, weak square
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Openly generated Boolean algebras and the Fodor-type reflection principle
Joint work with Sakaé Fuchino. Abstract: We prove that the Fodor-type Reflection Principle (FRP) is equivalent to the assertion that any Boolean algebra is openly generated if and only if it is $\aleph _2$-projective. Previously it was known that this … Continue reading
