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