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