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free Boolean algebra Shelah's Strong Hypothesis Ramsey theory over partitions P-Ideal Dichotomy Amenable C-sequence Subtle tree property Open Access regressive Souslin tree O-space Chromatic number Cohen real Successor of Singular Cardinal Non-saturation very good scale Iterated forcing Aronszajn tree diamond star Ulam matrix Strongly compact cardinal SNR square reflection principles Whitehead Problem tensor product graph Subadditive PFA stationary hitting Jonsson cardinal Analytic sets Diamond-sharp HOD coloring number higher Baire space Forcing with side conditions Monotonically far Cardinal Invariants Ostaszewski square Sakurai's Bell inequality Foundations Sigma-Prikry Partition Relations Martin's Axiom Diamond Reduced Power b-scale Entangled linear order C-sequence Commutative projection system Absoluteness Weakly compact cardinal L-space Strongly Luzin set Prikry-type forcing full tree Dowker space Kurepa Hypothesis PFA(S)[S] Rado's conjecture Square-Brackets Partition Relations Partition relations for trees Knaster and friends Greatly Mahlo Singular cardinals combinatorics Ineffable cardinal Almost Souslin Distributive tree Nonspecial tree incompactness Diamond for trees Lipschitz reduction Well-behaved magma Mandelbrot set 54G20 perfectly normal Almost-disjoint family Fast club Rainbow sets Intersection model Postprocessing function weak diamond xbox free Souslin tree Prevalent singular cardinals Uniformly coherent Erdos-Hajnal graphs Dushnik-Miller Slim tree Luzin set GMA Subnormal ideal Respecting tree Erdos Cardinal Fat stationary set Strong coloring Axiom R Small forcing polarized partition relation ccc AIM forcing Coherent tree OCA unbounded function weak square Chang's conjecture Uniformization Club Guessing stick countably metacompact Filter reflection stationary reflection super-Souslin tree Local Club Condensation. Generalized Clubs nonmeager set square principles Poset Large Cardinals sap indecomposable filter middle diamond S-Space Commutative cancellative semigroups Hedetniemi's conjecture Cardinal function club_AD Singular cofinality Reflecting stationary set strongly bounded groups weak Kurepa tree Almost countably chromatic Rock n' Roll Fodor-type reflection Microscopic Approach Uniformly homogeneous ZFC construction Hindman's Theorem approachability ideal Vanishing levels Countryman line transformations Souslin Tree projective Boolean algebra Universal Sequences Ascending path Hereditarily Lindelöf space Interval topology on trees Knaster Minimal Walks Successor of Regular Cardinal positive partition relation specializable Souslin tree Antichain Parameterized proxy principle Selective Ultrafilter Forcing Axioms Forcing Sierpinski's onto mapping principle Closed coloring Was Ulam right? Generalized descriptive set theory Ascent Path Subtle cardinal Singular Density Constructible Universe Precaliber
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
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
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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
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
17 Comments
Prikry Forcing
Recall that the chromatic number of a (symmetric) graph $(G,E)$, denoted $\text{Chr}(G,E)$, is the least (possible finite) cardinal $\kappa$, for which there exists a coloring $c:G\rightarrow\kappa$ such that $gEh$ entails $c(g)\neq c(h)$. Given a forcing notion $\mathbb P$, it is … Continue reading
Shelah’s approachability ideal (part 2)
In a previous post, we defined Shelah’s approachability ideal $I[\lambda]$. We remind the reader that a subset $S\subseteq\lambda$ is in $I[\lambda]$ iff there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$ such that for club many $\delta\in S$, the union … Continue reading
Posted in Blog, Expository, Open Problems
Tagged approachability ideal, Club Guessing
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