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