Archives
Keywords
projective Boolean algebra Ostaszewski square PFA Minimal Walks 54G20 perfectly normal PFA(S)[S] Singular cardinals combinatorics Rainbow sets Coherent tree Postprocessing function Jonsson cardinal L-space OCA Uniformly coherent Selective Ultrafilter Distributive tree free Boolean algebra indecomposable filter regressive Souslin tree Sierpinski's onto mapping principle Open Access Well-behaved magma Interval topology on trees Strongly compact cardinal Respecting tree weak Kurepa tree Almost countably chromatic Subtle cardinal Diamond free Souslin tree Axiom R Analytic sets C-sequence Ramsey theory over partitions b-scale stationary hitting Fast club tensor product graph Greatly Mahlo Dowker space Rado's conjecture Weakly compact cardinal Square-Brackets Partition Relations Monotonically far sap AIM forcing stationary reflection Generalized descriptive set theory Sakurai's Bell inequality Luzin set Intersection model Antichain S-Space Dushnik-Miller Sigma-Prikry Generalized Clubs Martin's Axiom Vanishing levels Erdos Cardinal Parameterized proxy principle Local Club Condensation. Fat stationary set Strong coloring Amenable C-sequence approachability ideal Strongly Luzin set Entangled linear order Forcing Axioms Ulam matrix reflection principles Subtle tree property Hindman's Theorem Large Cardinals Subadditive Lipschitz reduction coloring number Reduced Power Almost-disjoint family Aronszajn tree Forcing Almost Souslin Fodor-type reflection Nonspecial tree positive partition relation Precaliber Mandelbrot set Knaster Uniformly homogeneous Singular cofinality HOD unbounded function Souslin Tree Countryman line square Partition Relations Rock n' Roll Commutative cancellative semigroups super-Souslin tree club_AD stick Ineffable cardinal Hedetniemi's conjecture Cardinal function Hereditarily Lindelöf space Partition relations for trees Foundations nonmeager set Cardinal Invariants strongly bounded groups Non-saturation Closed coloring Prikry-type forcing Chang's conjecture Microscopic Approach Club Guessing Knaster and friends diamond star Commutative projection system Slim tree Universal Sequences higher Baire space P-Ideal Dichotomy Constructible Universe Diamond-sharp Poset ZFC construction Small forcing full tree polarized partition relation very good scale Successor of Regular Cardinal Singular Density Iterated forcing weak square Reflecting stationary set Forcing with side conditions Successor of Singular Cardinal middle diamond ccc Uniformization Chromatic number Kurepa Hypothesis Shelah's Strong Hypothesis incompactness square principles Diamond for trees Ascent Path O-space SNR Absoluteness transformations Erdos-Hajnal graphs xbox Filter reflection specializable Souslin tree Ascending path Prevalent singular cardinals weak diamond Subnormal ideal GMA Was Ulam right? Cohen real countably metacompact Whitehead Problem
Tag Archives: Cardinal Invariants
On the ideal J[kappa]
Abstract. Motivated by a question from a recent paper by Gilton, Levine and Stejskalova, we obtain a new characterization of the ideal $J[\kappa]$, from which we confirm that $\kappa$-Souslin trees exist in various models of interest. As a corollary we … Continue reading
Posted in Publications, Souslin Hypothesis
Tagged Cardinal Invariants, Cohen real, nonmeager set
1 Comment
Forcing with a Souslin tree makes $\mathfrak p=\omega_1$
I was meaning to include a proof of Farah’s lemma in my previous post, but then I realized that the slick proof assumes some background which may worth spelling out, first. Therefore, I am dedicating a short post for a … Continue reading
The S-space problem, and the cardinal invariant $\mathfrak p$
Recall that an $S$-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. Do they exist? Consistently, yes. However, Szentmiklóssy proved that compact $S$-spaces do not exist, assuming Martin’s Axiom. Pushing this further, Todorcevic later proved that … Continue reading
Posted in Blog, Expository, Open Problems
Tagged Cardinal Invariants, Hereditarily Lindelöf space, P-Ideal Dichotomy, PFA(S)[S], S-Space
4 Comments
Jones’ theorem on the cardinal invariant $\mathfrak p$
This post continues the study of the cardinal invariant $\mathfrak p$. We refer the reader to a previous post for all the needed background. For ordinals $\alpha,\alpha_0,\alpha_1,\beta,\beta_0,\beta_1$, the polarized partition relation $$\left(\begin{array}{c}\alpha\\\beta\end{array}\right)\rightarrow\left(\begin{array}{cc}\alpha_0&\alpha_1\\\beta_0&\beta_1\end{array}\right)$$ asserts that for every coloring $f:\alpha\times\beta\rightarrow 2$, (at least) … Continue reading
Bell’s theorem on the cardinal invariant $\mathfrak p$
In this post, we shall provide a proof to a famous theorem of Murray Bell stating that $MA_\kappa(\text{the class of }\sigma\text{-centered posets})$ holds iff $\kappa<\mathfrak p$. We commence with defining the cardinal invariant $\mathfrak p$. For sets $A$ and $B$, … Continue reading
Dushnik-Miller for regular cardinals (part 2)
In this post, we shall provide a proof of Todorcevic’s theorem, that $\mathfrak b=\omega_1$ implies $\omega_1\not\rightarrow(\omega_1,\omega+2)^2$. This will show that the Erdos-Rado theorem that we discussed in an earlier post, is consistently optimal. Our exposition of Todorcevic’s theorem would be … Continue reading
Infinite Combinatorial Topology
Back in 2005, as a master student, I attended a course by Boaz Tsaban, entitled “Infinite Combinatorial Topology”. A friend and I decided to produce lecture notes, but in a somewhat loose sense, that is: we sometimes omit material given … Continue reading
Posted in Notes
Tagged b-scale, Cardinal function, Cardinal Invariants, Hereditarily Lindelöf space
8 Comments
