Archives
Keywords
projective Boolean algebra super-Souslin tree ZFC construction Souslin Tree Strong coloring square Parameterized proxy principle Closed coloring Partition relations for trees b-scale Ramsey theory over partitions Strongly compact cardinal Postprocessing function C-sequence Diamond Countryman line Sakurai's Bell inequality Local Club Condensation. GMA Aronszajn tree Intersection model Generalized descriptive set theory Distributive tree Reduced Power Almost Souslin Club Guessing Subtle tree property Knaster Ulam matrix Prikry-type forcing nonmeager set Commutative projection system Well-behaved magma Almost countably chromatic Ascent Path Greatly Mahlo Successor of Singular Cardinal Universal Sequences sap Was Ulam right? Ascending path Shelah's Strong Hypothesis incompactness Fat stationary set Respecting tree diamond star polarized partition relation regressive Souslin tree Forcing with side conditions Forcing Axioms Rado's conjecture Successor of Regular Cardinal Subtle cardinal Cohen real full tree Analytic sets Whitehead Problem Rainbow sets Large Cardinals Singular Density Filter reflection stationary hitting Singular cardinals combinatorics higher Baire space Knaster and friends Foundations S-Space club_AD Amenable C-sequence Selective Ultrafilter PFA(S)[S] Ostaszewski square Prevalent singular cardinals 54G20 free Boolean algebra Commutative cancellative semigroups coloring number Minimal Walks weak Kurepa tree square principles Rock n' Roll Luzin set Nonspecial tree L-space Uniformly homogeneous Fodor-type reflection Erdos-Hajnal graphs Mandelbrot set unbounded function Sierpinski's onto mapping principle Chromatic number Martin's Axiom Almost-disjoint family Square-Brackets Partition Relations Poset Dushnik-Miller Entangled linear order reflection principles Microscopic Approach Absoluteness Non-saturation Precaliber stationary reflection perfectly normal Strongly Luzin set weak square countably metacompact Cardinal Invariants Weakly compact cardinal Partition Relations Vanishing levels Open Access very good scale SNR xbox Diamond-sharp Uniformly coherent Lipschitz reduction Diamond for trees stick weak diamond middle diamond Fast club approachability ideal Sigma-Prikry Chang's conjecture Erdos Cardinal Slim tree Monotonically far transformations Antichain HOD Ineffable cardinal O-space specializable Souslin tree Small forcing Reflecting stationary set Kurepa Hypothesis P-Ideal Dichotomy Forcing AIM forcing indecomposable filter Subadditive Uniformization strongly bounded groups Dowker space ccc Generalized Clubs Subnormal ideal Hereditarily Lindelöf space Coherent tree PFA OCA Singular cofinality Jonsson cardinal positive partition relation Hedetniemi's conjecture Constructible Universe Hindman's Theorem tensor product graph free Souslin tree Cardinal function Axiom R Interval topology on trees Iterated forcing
Author Archives: Assaf Rinot
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
Variations on diamond
Jensen’s diamond principle has many equivalent forms. The translation between these forms is often straight-forward, but there is one form whose equivalence to the usual form is somewhat surprising, and Devlin’s translation from one to the other, seems a little … Continue reading
The P-Ideal Dichotomy and the Souslin Hypothesis
John Krueger is visiting Toronto these days, and in a conversation today, we asked ourselves how do one prove the Abraham-Todorcevic theorem that PID implies SH. Namely, that the next statement implies that there are no Souslin trees: Definition. The … Continue reading
Infinite Combinatorics Seminar, Haifa University, June, 2012
I gave a talk at the University of Haifa on June 07, 2012, intended for general audience. Title: Strong Colorings: the study of the failure of generalized Ramsey statements Abstract: A strong coloring from $X$ to $Y$ is a function … Continue reading
Posted in Contributed Talks
Leave a comment
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
14 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
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
