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Amenable C-sequence Closed coloring Square-Brackets Partition Relations Successor of Singular Cardinal SNR Distributive tree Large Cardinals positive partition relation regressive Souslin tree Commutative cancellative semigroups reflection principles Diamond for trees Entangled linear order Strongly Luzin set Antichain Lipschitz reduction full tree Ascending path weak Kurepa tree club_AD Commutative projection system GMA Postprocessing function polarized partition relation weak square ZFC construction Absoluteness Reduced Power Mandelbrot set Dowker space Partition relations for trees sap Reflecting stationary set Subadditive Subtle cardinal Was Ulam right? ccc Cohen real Coherent tree Souslin Tree Nonspecial tree Selective Ultrafilter Ostaszewski square coloring number Open Access strongly bounded groups projective Boolean algebra Prevalent singular cardinals Hindman's Theorem Cardinal Invariants Fodor-type reflection tensor product graph Rado's conjecture Foundations AIM forcing square Respecting tree Knaster PFA Sakurai's Bell inequality Cardinal function unbounded function square principles Generalized descriptive set theory Uniformization Axiom R Uniformly homogeneous Filter reflection Fat stationary set Ulam matrix HOD Singular Density Erdos Cardinal Luzin set Aronszajn tree free Boolean algebra Diamond-sharp Local Club Condensation. S-Space higher Baire space super-Souslin tree Uniformly coherent O-space middle diamond Successor of Regular Cardinal Club Guessing stationary reflection L-space Generalized Clubs Almost countably chromatic Well-behaved magma Iterated forcing Almost Souslin Rainbow sets Non-saturation Countryman line Analytic sets Prikry-type forcing stationary hitting stick Sierpinski's onto mapping principle xbox Kurepa Hypothesis Singular cofinality OCA countably metacompact Hereditarily Lindelöf space Martin's Axiom PFA(S)[S] Erdos-Hajnal graphs Parameterized proxy principle Sigma-Prikry Partition Relations Intersection model nonmeager set Ramsey theory over partitions b-scale Poset Chang's conjecture Minimal Walks Strong coloring Universal Sequences Subtle tree property Constructible Universe Chromatic number Precaliber Knaster and friends Subnormal ideal Rock n' Roll approachability ideal weak diamond Forcing Ineffable cardinal Fast club indecomposable filter Whitehead Problem 54G20 transformations very good scale Shelah's Strong Hypothesis incompactness Hedetniemi's conjecture free Souslin tree Forcing Axioms specializable Souslin tree Greatly Mahlo C-sequence Dushnik-Miller Jonsson cardinal Monotonically far P-Ideal Dichotomy Microscopic Approach Diamond Slim tree Interval topology on trees Strongly compact cardinal Almost-disjoint family Small forcing Singular cardinals combinatorics perfectly normal Vanishing levels Forcing with side conditions diamond star Ascent Path Weakly compact cardinal
Author Archives: Assaf Rinot
Comparing rectangles with squares through rainbow sets
In Todorcevic’s class last week, he proved all the results of Chapter 8 from his Walks on Ordinals book, up to (and including) Theorem 8.1.11. The upshots are as follows: Every regular infinite cardinal $\theta$ admits a naturally defined function … Continue reading
ASL North American Meeting, March 2012
I gave a special session talk at the ASL 2012 North American Annual Meeting (Madison, March 31–April 3, 2012). Talk Title: The extent of the failure of Ramsey’s theorem at successor cardinals. Extended abstract: Ramsey’s theorem asserts that for every coloring … Continue reading
Posted in Invited Talks
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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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