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Successor of Regular Cardinal Dowker space transformations Open Access Ostaszewski square Strongly Luzin set Forcing Commutative cancellative semigroups Weakly compact cardinal Constructible Universe nonmeager set Forcing Axioms L-space Nonspecial tree Ulam matrix PFA(S)[S] Singular Density Square-Brackets Partition Relations Intersection model reflection principles Uniformly homogeneous strongly bounded groups Ineffable cardinal Martin's Axiom Parameterized proxy principle specializable Souslin tree positive partition relation Forcing with side conditions stick super-Souslin tree Almost countably chromatic Luzin set Commutative projection system weak Kurepa tree Fast club Reduced Power Minimal Walks Hedetniemi's conjecture HOD Closed coloring club_AD Antichain Prevalent singular cardinals Foundations perfectly normal Strong coloring Uniformly coherent Subnormal ideal diamond star Selective Ultrafilter Cohen real Slim tree Erdos Cardinal Jonsson cardinal square principles Knaster Interval topology on trees Ascent Path Fodor-type reflection Singular cardinals combinatorics P-Ideal Dichotomy Poset Prikry-type forcing Subadditive weak diamond Monotonically far polarized partition relation Local Club Condensation. square Sakurai's Bell inequality Countryman line Small forcing Non-saturation Successor of Singular Cardinal Large Cardinals Amenable C-sequence tensor product graph Kurepa Hypothesis Subtle tree property Greatly Mahlo Postprocessing function Ascending path Rock n' Roll Partition relations for trees Diamond Fat stationary set Chang's conjecture C-sequence Well-behaved magma Singular cofinality Diamond for trees Almost-disjoint family SNR Filter reflection coloring number ZFC construction higher Baire space Hereditarily Lindelöf space Generalized descriptive set theory Knaster and friends approachability ideal Shelah's Strong Hypothesis ccc Strongly compact cardinal Sierpinski's onto mapping principle Subtle cardinal Respecting tree Dushnik-Miller free Souslin tree Rainbow sets Distributive tree countably metacompact Chromatic number Was Ulam right? Reflecting stationary set Vanishing levels Almost Souslin projective Boolean algebra regressive Souslin tree Club Guessing S-Space Cardinal Invariants Axiom R Coherent tree free Boolean algebra Ramsey theory over partitions very good scale Partition Relations Lipschitz reduction Generalized Clubs Uniformization Rado's conjecture unbounded function Mandelbrot set sap Universal Sequences PFA xbox full tree Hindman's Theorem 54G20 GMA Analytic sets Absoluteness b-scale Precaliber Cardinal function stationary reflection stationary hitting Erdos-Hajnal graphs OCA Aronszajn tree AIM forcing weak square indecomposable filter Souslin Tree Entangled linear order O-space Diamond-sharp Whitehead Problem Sigma-Prikry Microscopic Approach Iterated forcing middle diamond incompactness
Author Archives: Assaf Rinot
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
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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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
The Engelking-Karlowicz theorem, and a useful corollary
Theorem (Engelking-Karlowicz, 1965). For cardinals $\kappa\le\lambda\le\mu\le 2^\lambda$, the following are equivalent: $\lambda^{<\kappa}=\lambda$; there exists a collection of functions, $\langle f_i:\mu\rightarrow\lambda\mid i<\lambda\rangle$, such that for every $X\in[\mu]^{<\kappa}$ and every function $f:X\rightarrow\lambda$, there exists some $i<\lambda$ with $f\subseteq f_i$. Proof. (2)$\Rightarrow$(1) Suppose … Continue reading
Kurepa trees and ineffable cardinals
Recall that $T$ is said to be a $\kappa$-Kurepa tree if $T$ is a tree of height $\kappa$, whose levels $T_\alpha$ has size $\le|\alpha|$ for co-boundedly many $\alpha<\kappa$, and such that the set of branches of $T$ has size $>\kappa$. … Continue reading
