Archives
Keywords
Well-behaved magma Uniformization Chang's conjecture Strongly Luzin set Diamond for trees Selective Ultrafilter free Boolean algebra transformations Ostaszewski square Local Club Condensation. Subadditive Antichain Ascent Path xbox Martin's Axiom incompactness Reflecting stationary set Open Access Poset C-sequence square Precaliber nonmeager set O-space weak Kurepa tree Cardinal function weak diamond unbounded function Hindman's Theorem Diamond Forcing Axioms Non-saturation Slim tree super-Souslin tree S-Space Sigma-Prikry higher Baire space Nonspecial tree Club Guessing Parameterized proxy principle full tree Filter reflection Greatly Mahlo Subtle tree property Almost Souslin indecomposable ultrafilter approachability ideal Singular cofinality projective Boolean algebra Rado's conjecture coloring number diamond star Rock n' Roll Prevalent singular cardinals countably metacompact sap Kurepa Hypothesis Generalized descriptive set theory Microscopic Approach Uniformly coherent stick Hedetniemi's conjecture Was Ulam right? Generalized Clubs Singular cardinals combinatorics Successor of Regular Cardinal Dushnik-Miller free Souslin tree Subtle cardinal Chromatic number Uniformly homogeneous Reduced Power Shelah's Strong Hypothesis Small forcing Ineffable cardinal Axiom R Respecting tree polarized partition relation club_AD Mandelbrot set Rainbow sets AIM forcing strongly bounded groups Large Cardinals Vanishing levels stationary hitting Closed coloring Diamond-sharp Fast club Prikry-type forcing L-space Forcing Dowker space stationary reflection specializable Souslin tree Absoluteness Sierpinski's onto mapping principle Jonsson cardinal Commutative projection system Cardinal Invariants Successor of Singular Cardinal Minimal Walks Intersection model Hereditarily Lindelöf space Universal Sequences weak square Foundations Erdos-Hajnal graphs ccc tensor product graph Lipschitz reduction Aronszajn tree Cohen real Postprocessing function SNR regressive Souslin tree Singular Density Strongly compact cardinal Almost countably chromatic Subnormal ideal Amenable C-sequence PFA Coherent tree Almost-disjoint family middle diamond Strong coloring Square-Brackets Partition Relations 54G20 b-scale square principles Luzin set Whitehead Problem positive partition relation Partition Relations Erdos Cardinal Distributive tree Souslin Tree Fat stationary set reflection principles Weakly compact cardinal ZFC construction HOD Countryman line OCA Fodor-type reflection Constructible Universe Knaster Ulam matrix Commutative cancellative semigroups very good scale Knaster and friends P-Ideal Dichotomy Sakurai's Bell inequality Analytic sets Ramsey theory over partitions GMA PFA(S)[S] Iterated forcing
Category Archives: Blog
Afghan Whigs on Jimmy Fallon
Performing “I’m Her Slave” (from their album Congregation) at NBC’s studios, 22-May-2012:
The chromatic numbers of the Erdos-Hajnal graphs
Recall that a coloring $c:G\rightarrow\kappa$ of an (undirected) graph $(G,E)$ is said to be chromatic if $c(v_1)\neq c(v_2)$ whenever $\{v_1,v_2\}\in E$. Then, the chromatic number of a graph $(G,E)$ is the least cardinal $\kappa$ for which there exists a chromatic … Continue reading
Posted in Blog, Expository
Tagged Chromatic number, Erdos-Hajnal graphs, Rado's conjecture, reflection principles
13 Comments
Shelah’s approachability ideal (part 1)
Given an infinite cardinal $\lambda$, Shelah defines an ideal $I[\lambda]$ as follows. Definition (Shelah, implicit in here). A set $S$ is in $I[\lambda]$ iff $S\subseteq\lambda$ and there exists a collection $\{ \mathcal D_\alpha\mid\alpha<\lambda\}\subseteq\mathcal [\mathcal P(\lambda)]^{<\lambda}$, and some club $E\subseteq\lambda$, so … Continue reading
Review: Is classical set theory compatible with quantum experiments?
Yesterday, I attended a talk at the Quantum Foundations seminar at the beautiful Perimeter Institute for Theoretical Physics (Waterloo, Ontario). The (somewhat provocative) title of the talk was “Is Classical Set Theory Compatible with Quantum Experiments?”, and the speaker was Radu … 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
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
