Archives
Keywords
square middle diamond Was Ulam right? Dushnik-Miller xbox Universal Sequences Erdos Cardinal Foundations stick O-space Singular Density Jonsson cardinal Lipschitz reduction Absoluteness Chang's conjecture Dowker space unbounded function Ineffable cardinal Rainbow sets Local Club Condensation. reflection principles specializable Souslin tree Subnormal ideal Souslin Tree positive partition relation Strong coloring transformations Parameterized proxy principle Forcing with side conditions Precaliber Analytic sets strongly bounded groups Closed coloring perfectly normal Cardinal function L-space Cohen real Forcing Axioms Entangled linear order S-Space Reduced Power incompactness Kurepa Hypothesis Strongly Luzin set Erdos-Hajnal graphs ccc Luzin set Singular cardinals combinatorics Sierpinski's onto mapping principle AIM forcing super-Souslin tree Interval topology on trees Martin's Axiom Reflecting stationary set Prevalent singular cardinals Uniformly coherent Almost-disjoint family Minimal Walks Constructible Universe Ramsey theory over partitions approachability ideal Small forcing Respecting tree Club Guessing Greatly Mahlo Fat stationary set diamond star Coherent tree very good scale OCA Rado's conjecture Prikry-type forcing Mandelbrot set Singular cofinality Iterated forcing projective Boolean algebra Successor of Singular Cardinal Ulam matrix Diamond for trees Slim tree Hereditarily Lindelöf space Partition Relations Diamond regressive Souslin tree Postprocessing function Knaster and friends SNR Monotonically far Almost Souslin countably metacompact ZFC construction Distributive tree weak diamond Axiom R weak square Antichain stationary hitting Aronszajn tree Filter reflection Cardinal Invariants Diamond-sharp sap Selective Ultrafilter Fodor-type reflection Partition relations for trees Chromatic number Large Cardinals Open Access Amenable C-sequence Microscopic Approach PFA(S)[S] Strongly compact cardinal b-scale Subtle cardinal Successor of Regular Cardinal Vanishing levels Hindman's Theorem P-Ideal Dichotomy Countryman line higher Baire space Commutative projection system Almost countably chromatic free Boolean algebra free Souslin tree weak Kurepa tree Sigma-Prikry Ascent Path Forcing Uniformization Subadditive Generalized descriptive set theory full tree Non-saturation indecomposable filter coloring number Well-behaved magma stationary reflection polarized partition relation HOD Whitehead Problem Nonspecial tree Commutative cancellative semigroups PFA Weakly compact cardinal Hedetniemi's conjecture C-sequence Sakurai's Bell inequality Subtle tree property Rock n' Roll nonmeager set Poset Knaster tensor product graph square principles club_AD Uniformly homogeneous Fast club Square-Brackets Partition Relations GMA 54G20 Ascending path Shelah's Strong Hypothesis Intersection model Ostaszewski square Generalized Clubs
Category Archives: Publications
Hedetniemi’s conjecture for uncountable graphs
Abstract. It is proved that in Godel’s constructible universe, for every successor cardinal $\kappa$, there exist graphs $\mathcal G$ and $\mathcal H$ of size and chromatic number $\kappa$, for which the tensor product graph $\mathcal G\times\mathcal H$ is countably chromatic. … Continue reading
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
Rectangular square-bracket operation for successor of regular cardinals
Joint work with Stevo Todorcevic. Extended Abstract: Consider the coloring statement $\lambda^+\nrightarrow[\lambda^+;\lambda^+]^2_{\lambda^+}$ for a given regular cardinal $\lambda$: In 1990, Shelah proved the above for $\lambda>2^{\aleph_0}$; In 1991, Shelah proved the above for $\lambda>\aleph_1$; In 1997, Shelah proved the above … Continue reading
The search for diamonds
Abstract: This is a review I wrote for the Bulletin of Symbolic Logic on the following papers: Saharon Shelah, Middle Diamond, Archive for Mathematical Logic, vol. 44 (2005), pp. 527–560. Saharon Shelah, Diamonds, Proceedings of the American Mathematical Society, vol. … Continue reading
Posted in Publications, Reviews, Squares and Diamonds
Tagged Diamond, middle diamond, weak diamond, weak square
1 Comment
Aspects of singular cofinality
Abstract. We study properties of closure operators of singular cofinality, and introduce several ZFC sufficient and equivalent conditions for the existence of antichain sequences in posets of singular cofinality. We also notice that the Proper Forcing Axiom implies the Milner-Sauer … Continue reading
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
Leave a comment
On topological spaces of singular density and minimal weight
Abstract: We introduce a weakening of the Generalized Continuum Hypothesis, which we will refer to as the Prevalent Singular cardinals Hypothesis (PSH), and show it implies that every topological space of density and weight $\aleph_{\omega_1}$ is not hereditarily Lindelöf. The assumption … Continue reading
A topological reflection principle equivalent to Shelah’s strong hypothesis
Abstract: We notice that Shelah’s Strong Hypothesis (SSH) is equivalent to the following reflection principle: Suppose $\mathbb X$ is an (infinite) first-countable space whose density is a regular cardinal, $\kappa$. If every separable subspace of $\mathbb X$ is of cardinality at most … Continue reading
Posted in Compactness, Publications, Topology
Tagged 03E04, 03E65, 54G15, Open Access, Shelah's Strong Hypothesis
Comments Off on A topological reflection principle equivalent to Shelah’s strong hypothesis
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
Posted in Compactness, Publications
Tagged 03E35, 03E55, 03E65, 03E75, 03G05, 06E05, Axiom R, Fodor-type reflection, free Boolean algebra, projective Boolean algebra, Shelah's Strong Hypothesis, stationary reflection
Comments Off on Openly generated Boolean algebras and the Fodor-type reflection principle
