Archives
Keywords
square principles specializable Souslin tree PFA(S)[S] weak diamond Cardinal Invariants Chromatic number approachability ideal Universal Sequences Ulam matrix Fodor-type reflection Ascending path C-sequence middle diamond b-scale diamond star Local Club Condensation. Greatly Mahlo Coherent tree 54G20 Commutative cancellative semigroups Selective Ultrafilter Square-Brackets Partition Relations Interval topology on trees AIM forcing Successor of Singular Cardinal Singular Density Respecting tree Subnormal ideal Whitehead Problem Strongly Luzin set Martin's Axiom Kurepa Hypothesis Uniformly homogeneous Filter reflection strongly bounded groups Countryman line Precaliber Forcing Aronszajn tree Generalized descriptive set theory Subtle tree property regressive Souslin tree Foundations xbox free Boolean algebra Strong coloring sap Fat stationary set O-space stationary reflection Generalized Clubs Rado's conjecture Reduced Power Minimal Walks Amenable C-sequence tensor product graph Absoluteness Hindman's Theorem Nonspecial tree Chang's conjecture Strongly compact cardinal Vanishing levels Uniformly coherent very good scale Closed coloring Souslin Tree Open Access Entangled linear order Monotonically far Singular cofinality Ascent Path coloring number Dowker space free Souslin tree Luzin set Club Guessing P-Ideal Dichotomy Partition relations for trees Singular cardinals combinatorics Knaster and friends HOD Cardinal function S-Space incompactness Subadditive Large Cardinals reflection principles positive partition relation Partition Relations Small forcing Weakly compact cardinal Distributive tree Lipschitz reduction ccc Commutative projection system Forcing with side conditions Well-behaved magma SNR Knaster Sigma-Prikry Reflecting stationary set Non-saturation Diamond-sharp super-Souslin tree Jonsson cardinal Successor of Regular Cardinal Iterated forcing Almost countably chromatic Diamond for trees Prevalent singular cardinals Hedetniemi's conjecture Poset L-space full tree Erdos-Hajnal graphs Axiom R Hereditarily Lindelöf space Was Ulam right? Almost-disjoint family PFA polarized partition relation Diamond nonmeager set Erdos Cardinal Forcing Axioms perfectly normal GMA Ineffable cardinal stick Parameterized proxy principle Microscopic Approach square Ramsey theory over partitions Sierpinski's onto mapping principle Rock n' Roll Prikry-type forcing projective Boolean algebra Shelah's Strong Hypothesis Sakurai's Bell inequality transformations Cohen real countably metacompact unbounded function Intersection model indecomposable filter OCA club_AD ZFC construction Constructible Universe Rainbow sets stationary hitting Uniformization weak Kurepa tree higher Baire space Antichain Almost Souslin Dushnik-Miller Ostaszewski square Mandelbrot set Analytic sets Subtle cardinal Postprocessing function Fast club Slim tree weak square
Author Archives: Assaf Rinot
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
Leave a comment
The uniformization property for $\aleph_2$
Given a subset of a regular uncountable cardinal $S\subseteq\kappa$, $UP_S$ (read: “the uniformization property holds for $S$”) asserts that for every sequence $\overrightarrow f=\langle f_\alpha\mid \alpha\in S\rangle$ satisfying for all $\alpha\in S$: $f_\alpha$ is a 2-valued function; $\text{dom}(f_\alpha)$ is a … Continue reading
