Archives
Keywords
Subtle cardinal Club Guessing Diamond projective Boolean algebra Slim tree Prikry-type forcing coloring number Fodor-type reflection Intersection model Generalized descriptive set theory Axiom R Mandelbrot set Fat stationary set Ulam matrix diamond star perfectly normal ZFC construction SNR Cardinal Invariants Microscopic Approach Well-behaved magma Cardinal function Prevalent singular cardinals Coherent tree Knaster Selective Ultrafilter Filter reflection Vanishing levels transformations Sigma-Prikry Diamond for trees polarized partition relation regressive Souslin tree Singular cardinals combinatorics square principles Amenable C-sequence Minimal Walks Constructible Universe Dowker space positive partition relation Square-Brackets Partition Relations Jonsson cardinal Precaliber Sierpinski's onto mapping principle Distributive tree weak diamond OCA Open Access free Boolean algebra square Closed coloring P-Ideal Dichotomy Strongly Luzin set stationary hitting Almost-disjoint family Knaster and friends Uniformly homogeneous Local Club Condensation. Reflecting stationary set Large Cardinals xbox Iterated forcing Postprocessing function Erdos-Hajnal graphs Chang's conjecture AIM forcing Rado's conjecture Almost countably chromatic 54G20 Partition relations for trees Analytic sets full tree Foundations Subnormal ideal incompactness Nonspecial tree L-space O-space higher Baire space stationary reflection Chromatic number stick countably metacompact Aronszajn tree reflection principles Almost Souslin Antichain Kurepa Hypothesis Reduced Power approachability ideal Forcing with side conditions specializable Souslin tree PFA Uniformization Entangled linear order Whitehead Problem free Souslin tree Absoluteness Rock n' Roll middle diamond GMA Countryman line Strong coloring Greatly Mahlo Hindman's Theorem Hereditarily Lindelöf space indecomposable filter unbounded function Non-saturation Subtle tree property Lipschitz reduction Universal Sequences PFA(S)[S] Ineffable cardinal ccc Partition Relations Successor of Regular Cardinal very good scale weak Kurepa tree Weakly compact cardinal Was Ulam right? Respecting tree HOD Commutative projection system C-sequence Martin's Axiom Luzin set club_AD Parameterized proxy principle Interval topology on trees Forcing Axioms Erdos Cardinal Small forcing Strongly compact cardinal Poset Ascent Path weak square Generalized Clubs Rainbow sets tensor product graph Souslin Tree Diamond-sharp Singular Density Fast club Ostaszewski square b-scale Cohen real Monotonically far nonmeager set Uniformly coherent super-Souslin tree Dushnik-Miller Ascending path Commutative cancellative semigroups S-Space Forcing Sakurai's Bell inequality sap Singular cofinality Ramsey theory over partitions Shelah's Strong Hypothesis Hedetniemi's conjecture Subadditive Successor of Singular Cardinal strongly bounded groups
Author Archives: Assaf Rinot
Rectangular square-bracket operation for successor of regular cardinals
Joint work with Stevo Todorcevic. Extended Abstract: Consider the coloring statement $\lambda^+\nrightarrow[\lambda^+;\lambda^+]^2_{\lambda^+}$ for a given regular cardinal $\lambda$: In 1990, Shelah proved the above for $\lambda>2^{\aleph_0}$; In 1991, Shelah proved the above for $\lambda>\aleph_1$; In 1997, Shelah proved the above … Continue reading
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
2 Comments
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
