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Diamond Singular cofinality Entangled linear order projective Boolean algebra Minimal Walks Ostaszewski square specializable Souslin tree Cardinal function Martin's Axiom stick perfectly normal coloring number Absoluteness PFA S-Space Hindman's Theorem Rock n' Roll Generalized descriptive set theory Non-saturation Club Guessing weak Kurepa tree Prikry-type forcing Postprocessing function Countryman line Luzin set unbounded function transformations OCA full tree Vanishing levels Hereditarily Lindelöf space ZFC construction Local Club Condensation. Intersection model Cardinal Invariants Ascending path Erdos Cardinal Singular Density Coherent tree Cohen real Nonspecial tree Selective Ultrafilter Uniformly coherent Reflecting stationary set AIM forcing Souslin Tree Rado's conjecture Fat stationary set Monotonically far Filter reflection Axiom R Strong coloring Well-behaved magma Sakurai's Bell inequality Mandelbrot set reflection principles SNR Precaliber Ramsey theory over partitions Dowker space xbox b-scale free Souslin tree Successor of Regular Cardinal Uniformization Large Cardinals square Whitehead Problem Subadditive Subnormal ideal O-space Aronszajn tree Forcing Lipschitz reduction approachability ideal Hedetniemi's conjecture Small forcing stationary reflection Reduced Power ccc Almost Souslin Successor of Singular Cardinal Kurepa Hypothesis Respecting tree GMA Almost-disjoint family middle diamond Closed coloring free Boolean algebra Diamond for trees Rainbow sets Fast club HOD Commutative cancellative semigroups Slim tree Knaster Amenable C-sequence Partition relations for trees Distributive tree Open Access Subtle tree property Shelah's Strong Hypothesis Partition Relations Strongly Luzin set PFA(S)[S] Was Ulam right? Microscopic Approach Forcing Axioms Foundations Chang's conjecture Analytic sets weak square Parameterized proxy principle Uniformly homogeneous Subtle cardinal stationary hitting Sigma-Prikry Diamond-sharp P-Ideal Dichotomy Constructible Universe countably metacompact Weakly compact cardinal L-space Generalized Clubs polarized partition relation square principles club_AD Commutative projection system weak diamond Knaster and friends 54G20 Almost countably chromatic tensor product graph Strongly compact cardinal Iterated forcing Ulam matrix Square-Brackets Partition Relations C-sequence Antichain Dushnik-Miller Prevalent singular cardinals Chromatic number positive partition relation Universal Sequences Ineffable cardinal Jonsson cardinal incompactness higher Baire space indecomposable filter Poset diamond star Sierpinski's onto mapping principle Greatly Mahlo sap strongly bounded groups Erdos-Hajnal graphs super-Souslin tree Forcing with side conditions very good scale Fodor-type reflection Singular cardinals combinatorics nonmeager set regressive Souslin tree Interval topology on trees Ascent Path
Category Archives: Expository
A Kurepa tree from diamond-plus
Recall that $T$ is said to be a $\kappa$-Kurepa tree if $T$ is a tree of height $\kappa$, whose levels $T_\alpha$ has size $\le|\alpha|$ for co-boundedly many $\alpha<\kappa$, and such that the set of branches of $T$ has size $>\kappa$. … Continue reading
Posted in Blog, Expository
Tagged diamond star, Kurepa Hypothesis
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The S-space problem, and the cardinal invariant $\mathfrak b$
Recall that an S-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In a previous post, we showed that such a space exists after adding a Cohen real. Here, we shall construct one from an arithmetic … Continue reading
An $S$-space from a Cohen real
Recall that an $S$-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In this post, we shall establish the consistency of the existence of such a space. Theorem (Roitman, 1979). Let $\mathbb C=({}^{<\omega}\omega,\subseteq)$ be the notion of … Continue reading
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
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 Hereditarily Lindelöf space, P-Ideal Dichotomy, PFA(S)[S], S-Space
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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
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
Posted in Blog, Expository
Tagged polarized partition relation, Sierpinski's onto mapping principle
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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
The $\Delta$-system lemma: an elementary proof
Here is an elementary proof of (the finitary version of) the $\Delta$-system lemma. Thanks goes to Bill Weiss who showed me this proof! Lemma. Suppose that $\kappa$ is a regular uncountable cardinal, and $\mathcal A$ is a $\kappa$-sized family of finite … Continue reading
Posted in Blog, Expository, Surprisingly short
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