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Singular cofinality ccc Postprocessing function specializable Souslin tree Diamond tensor product graph Ascent Path Coherent tree Weakly compact cardinal Ramsey theory over partitions Antichain Reduced Power Singular cardinals combinatorics weak Kurepa tree Ostaszewski square Well-behaved magma Strongly Luzin set Square-Brackets Partition Relations Luzin set Erdos-Hajnal graphs free Souslin tree reflection principles Analytic sets b-scale Commutative cancellative semigroups Jonsson cardinal Iterated forcing Respecting tree SNR stationary hitting Souslin Tree coloring number polarized partition relation Forcing Mandelbrot set Subtle tree property Interval topology on trees Knaster Commutative projection system S-Space Ineffable cardinal Monotonically far Axiom R Whitehead Problem Reflecting stationary set Nonspecial tree Small forcing Subtle cardinal Uniformly homogeneous Aronszajn tree Cardinal function Diamond for trees Filter reflection Ascending path Partition Relations Entangled linear order GMA positive partition relation Strong coloring regressive Souslin tree Slim tree Cardinal Invariants Generalized Clubs perfectly normal PFA Foundations Intersection model higher Baire space very good scale Minimal Walks Almost countably chromatic Fodor-type reflection P-Ideal Dichotomy Dowker space Hindman's Theorem Uniformization 54G20 Amenable C-sequence Was Ulam right? Successor of Regular Cardinal Rado's conjecture middle diamond club_AD Shelah's Strong Hypothesis Dushnik-Miller unbounded function PFA(S)[S] Almost Souslin weak diamond OCA Subadditive Lipschitz reduction incompactness full tree Non-saturation Universal Sequences transformations Vanishing levels Closed coloring Precaliber Sierpinski's onto mapping principle Microscopic Approach Knaster and friends Uniformly coherent Distributive tree Prevalent singular cardinals Cohen real free Boolean algebra Open Access Fat stationary set indecomposable filter projective Boolean algebra Successor of Singular Cardinal approachability ideal countably metacompact Countryman line O-space Diamond-sharp Hedetniemi's conjecture Sakurai's Bell inequality Poset xbox Sigma-Prikry nonmeager set Fast club Partition relations for trees Greatly Mahlo diamond star Club Guessing Rainbow sets Forcing Axioms Subnormal ideal Forcing with side conditions C-sequence Chromatic number Ulam matrix square principles stick Local Club Condensation. Rock n' Roll ZFC construction L-space AIM forcing Prikry-type forcing Martin's Axiom Constructible Universe Parameterized proxy principle Almost-disjoint family Erdos Cardinal stationary reflection Chang's conjecture Singular Density square Hereditarily Lindelöf space Kurepa Hypothesis Absoluteness Large Cardinals HOD super-Souslin tree sap Generalized descriptive set theory Selective Ultrafilter weak square strongly bounded groups Strongly compact cardinal
Author Archives: Assaf Rinot
Pure logic
While traveling downtown today, I came across a sign near a local church, with a quotation of Saint-Exupéry:
Jane’s Addiction visiting Toronto
Last night, I went to see a live show by Jane’s Addiction, in downtown Toronto. Here’s a video snippet from that show which I could found on YouTube: The playlist was excellent, but there was one song which I was … Continue reading
c.c.c. vs. the Knaster property
After my previous post on Mekler’s characterization of c.c.c. notions of forcing, Sam, Mike and myself discussed the value of it . We noticed that a prevalent verification of the c.c.c. goes like this: given an uncountable set of conditions, … Continue reading
Dushnik-Miller for regular cardinals (part 3)
Here is what we already know about the Dushnik-Miller theorem in the case of $\omega_1$ (given our earlier posts on the subject): $\omega_1\rightarrow(\omega_1,\omega+1)^2$ holds in ZFC; $\omega_1\rightarrow(\omega_1,\omega+2)^2$ may consistently fail; $\omega_1\rightarrow(\omega_1,\omega_1)^2$ fails in ZFC. In this post, we shall provide … Continue reading
A large cardinal in the constructible universe
In this post, we shall provide a proof of Silver’s theorem that the Erdos caridnal $\kappa(\omega)$ relativizes to Godel’s constructible universe. First, recall some definitions. Given a function $f:[\kappa]^{<\omega}\rightarrow \mu$, we say that $I\subseteq\kappa$ is a set of indiscernibles for … Continue reading
An inconsistent form of club guessing
In this post, we shall present an answer (due to P. Larson) to a question by A. Primavesi concerning a certain strong form of club guessing. We commence with recalling Shelah’s concept of club guessing. Concept (Shelah). Given a regular … Continue reading
c.c.c. forcing without combinatorics
In this post, we shall discuss a short paper by Alan Mekler from 1984, concerning a non-combinatorial verification of the c.c.c. property for forcing notions. Recall that a notion of forcing $\mathbb P$ is said to satisfy the c.c.c. iff … Continue reading
Dushnik-Miller for singular cardinals (part 2)
In the first post on this subject, we provided a proof of $\lambda\rightarrow(\lambda,\omega+1)^2$ for every regular uncountable cardinal $\lambda$. In the second post, we provided a proof of $\lambda\rightarrow(\lambda,\omega)^2$ for every singular cardinal $\lambda$, and showed that $\lambda\rightarrow(\lambda,\omega+1)^2$ fails for every … Continue reading
Posted in Blog, Expository
Tagged Dushnik-Miller, Partition Relations, Singular cardinals combinatorics
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Dushnik-Miller for regular cardinals (part 2)
In this post, we shall provide a proof of Todorcevic’s theorem, that $\mathfrak b=\omega_1$ implies $\omega_1\not\rightarrow(\omega_1,\omega+2)^2$. This will show that the Erdos-Rado theorem that we discussed in an earlier post, is consistently optimal. Our exposition of Todorcevic’s theorem would be … Continue reading
Dushnik-Miller for singular cardinals (part 1)
Continuing the previous post, let us now prove the following. Theorem (Erdos-Dushnik-Miller, 1941). For every singular cardinal λ, we have: $$\lambda\rightarrow(\lambda,\omega)^2.$$ Proof. Suppose that $\lambda$ is a singular cardinal, and $c:[\lambda]^2\rightarrow\{0,1\}$ is a given coloring. For any ordinal $\alpha<\lambda$, denote … Continue reading
