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weak Kurepa tree Almost Souslin Fodor-type reflection countably metacompact Iterated forcing Ascent Path Erdos-Hajnal graphs regressive Souslin tree Club Guessing Knaster and friends weak diamond Chromatic number Slim tree Strongly Luzin set Constructible Universe Selective Ultrafilter Ulam matrix Generalized Clubs Prevalent singular cardinals Foundations Coherent tree Vanishing levels Successor of Regular Cardinal Fat stationary set S-Space Whitehead Problem Forcing nonmeager set Singular Density Almost-disjoint family Chang's conjecture Greatly Mahlo stick Successor of Singular Cardinal Local Club Condensation. Weakly compact cardinal Forcing Axioms Poset Ostaszewski square Universal Sequences Diamond-sharp higher Baire space weak square Large Cardinals coloring number L-space polarized partition relation Reduced Power Uniformization Strongly compact cardinal Martin's Axiom Nonspecial tree Antichain Microscopic Approach Square-Brackets Partition Relations PFA(S)[S] Rado's conjecture Luzin set transformations diamond star Mandelbrot set Filter reflection PFA GMA Kurepa Hypothesis P-Ideal Dichotomy Postprocessing function incompactness Absoluteness Commutative projection system Prikry-type forcing Precaliber Uniformly coherent Subtle tree property positive partition relation very good scale square principles Generalized descriptive set theory Countryman line b-scale club_AD Ineffable cardinal tensor product graph Sakurai's Bell inequality AIM forcing OCA projective Boolean algebra free Souslin tree ZFC construction SNR Souslin Tree Diamond for trees Hereditarily Lindelöf space Amenable C-sequence Erdos Cardinal Rainbow sets Minimal Walks C-sequence Aronszajn tree Rock n' Roll super-Souslin tree Hedetniemi's conjecture Reflecting stationary set Singular cardinals combinatorics Dushnik-Miller Partition Relations Axiom R indecomposable ultrafilter xbox strongly bounded groups Dowker space Fast club Singular cofinality Sigma-Prikry specializable Souslin tree Cardinal Invariants sap middle diamond stationary reflection Lipschitz reduction Closed coloring Almost countably chromatic Distributive tree Non-saturation Shelah's Strong Hypothesis Sierpinski's onto mapping principle approachability ideal Respecting tree stationary hitting Subadditive Intersection model HOD Jonsson cardinal Parameterized proxy principle Subnormal ideal Diamond Uniformly homogeneous full tree Knaster square Hindman's Theorem O-space Commutative cancellative semigroups Open Access reflection principles Strong coloring free Boolean algebra unbounded function Small forcing Subtle cardinal Was Ulam right Cardinal function Ramsey theory over partitions Analytic sets ccc Cohen real Well-behaved magma 54G20
Category Archives: Blog
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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Erdős 100
The influential mathematician Paul Erdős was born 100 years ago, 26 March 1913, in Budapest. One evidence of his impact on mathematics is reflected in the particular list of invited speakers for the upcoming conference in his honor. Erdős is also … 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
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
A natural Mandelbrot set
Chris Hadfield is a Canadian astronaut, with a very high-profile twitter account. He posts there beautiful photos everyday, and I (plus half a million followers) enjoy it very much. Today, Chris posted the following picture: and I find it quite … Continue reading
