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