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Nonspecial tree Sigma-Prikry Selective Ultrafilter Axiom R 54G20 GMA Intersection model Successor of Singular Cardinal stationary hitting Local Club Condensation. strongly bounded groups Singular cofinality Foundations higher Baire space Mandelbrot set Subadditive stationary reflection Non-saturation Forcing middle diamond L-space Uniformly homogeneous Subnormal ideal Jonsson cardinal S-Space regressive Souslin tree Was Ulam right? Singular cardinals combinatorics Uniformization unbounded function super-Souslin tree Analytic sets Diamond-sharp Commutative projection system Strongly Luzin set Generalized descriptive set theory Fast club Small forcing Ramsey theory over partitions Diamond for trees Chang's conjecture free Souslin tree Knaster Fat stationary set Lipschitz reduction free Boolean algebra Greatly Mahlo Antichain countably metacompact Monotonically far Prevalent singular cardinals AIM forcing diamond star Dowker space tensor product graph transformations Hereditarily Lindelöf space Rock n' Roll Forcing Axioms perfectly normal Whitehead Problem Absoluteness Large Cardinals indecomposable filter Respecting tree Knaster and friends Countryman line Ascending path Erdos Cardinal Fodor-type reflection full tree Subtle tree property Cohen real Filter reflection Rainbow sets specializable Souslin tree Ostaszewski square Erdos-Hajnal graphs positive partition relation Distributive tree Poset Rado's conjecture Reduced Power Subtle cardinal Reflecting stationary set Dushnik-Miller Sierpinski's onto mapping principle Strongly compact cardinal Precaliber Cardinal function Club Guessing Successor of Regular Cardinal ccc very good scale reflection principles ZFC construction Hindman's Theorem Weakly compact cardinal Microscopic Approach Entangled linear order Sakurai's Bell inequality Almost Souslin stick P-Ideal Dichotomy Almost countably chromatic Chromatic number Amenable C-sequence square approachability ideal OCA Cardinal Invariants HOD Postprocessing function Universal Sequences C-sequence Forcing with side conditions xbox nonmeager set Well-behaved magma Parameterized proxy principle Strong coloring b-scale Generalized Clubs Shelah's Strong Hypothesis Open Access Uniformly coherent Commutative cancellative semigroups Almost-disjoint family Partition Relations SNR Souslin Tree Coherent tree Hedetniemi's conjecture Square-Brackets Partition Relations club_AD O-space Vanishing levels Slim tree projective Boolean algebra coloring number polarized partition relation weak diamond Ineffable cardinal Closed coloring Constructible Universe Interval topology on trees weak square incompactness Ascent Path square principles Martin's Axiom Minimal Walks Luzin set Kurepa Hypothesis Singular Density Prikry-type forcing Ulam matrix PFA Partition relations for trees Aronszajn tree sap PFA(S)[S] Iterated forcing weak Kurepa tree Diamond
Category Archives: Blog
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
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
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
What’s next?
I took an offer for a tenure-track position at the Mathematics department of Bar-Ilan University.
Posted in Blog
13 Comments
Review: Stevo Todorcevic’s CRM-Fields-PIMS Prize Lecture
After winning the 2012 CRM-Fields-PIMS Prize, Stevo Todorcevic gave a series of talks on his research: at CRM, at PIMS and at the Fields Institute. The director of the Fields Institute asked me to write a short review on Stevo’s … Continue reading
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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