Archives
Keywords
Greatly Mahlo Respecting tree Constructible Universe stick Large Cardinals Dowker space Strong coloring Aronszajn tree Iterated forcing Cohen real Kurepa Hypothesis countably metacompact Axiom R Poset stationary reflection Forcing Axioms middle diamond Dushnik-Miller SNR Minimal Walks Forcing with side conditions Filter reflection Intersection model Parameterized proxy principle weak diamond Interval topology on trees Erdos Cardinal PFA Uniformly homogeneous club_AD Small forcing Weakly compact cardinal Sierpinski's onto mapping principle Singular Density GMA Hindman's Theorem higher Baire space Singular cardinals combinatorics Commutative cancellative semigroups incompactness Knaster and friends Hedetniemi's conjecture Countryman line xbox Almost countably chromatic perfectly normal Diamond for trees tensor product graph full tree positive partition relation HOD Uniformization Rainbow sets Rock n' Roll L-space Cardinal Invariants Subnormal ideal Reflecting stationary set Hereditarily Lindelöf space Subadditive Generalized Clubs b-scale coloring number regressive Souslin tree Fast club specializable Souslin tree reflection principles P-Ideal Dichotomy 54G20 AIM forcing Chromatic number Universal Sequences free Boolean algebra Ineffable cardinal stationary hitting nonmeager set Erdos-Hajnal graphs Strongly compact cardinal sap polarized partition relation Ascent Path Subtle cardinal Chang's conjecture Amenable C-sequence Jonsson cardinal indecomposable filter Forcing Absoluteness super-Souslin tree Precaliber Prevalent singular cardinals Coherent tree Fodor-type reflection Singular cofinality Knaster Non-saturation Foundations approachability ideal Cardinal function O-space free Souslin tree diamond star Well-behaved magma Reduced Power projective Boolean algebra Selective Ultrafilter Diamond Luzin set Mandelbrot set Analytic sets Almost Souslin Was Ulam right? Closed coloring Open Access square very good scale C-sequence Microscopic Approach Diamond-sharp S-Space Uniformly coherent Successor of Singular Cardinal Shelah's Strong Hypothesis Lipschitz reduction Fat stationary set ccc PFA(S)[S] Commutative projection system Ascending path Antichain Rado's conjecture Prikry-type forcing Ulam matrix Martin's Axiom Partition Relations Whitehead Problem weak Kurepa tree Partition relations for trees Strongly Luzin set strongly bounded groups Postprocessing function Almost-disjoint family transformations Ostaszewski square Subtle tree property Local Club Condensation. Entangled linear order ZFC construction Sakurai's Bell inequality Slim tree weak square OCA square principles Square-Brackets Partition Relations Sigma-Prikry Club Guessing Nonspecial tree Vanishing levels Souslin Tree unbounded function Ramsey theory over partitions Monotonically far Successor of Regular Cardinal Generalized descriptive set theory Distributive tree
Author Archives: Assaf Rinot
The uniformization property for $\aleph_2$
Given a subset of a regular uncountable cardinal $S\subseteq\kappa$, $UP_S$ (read: “the uniformization property holds for $S$”) asserts that for every sequence $\overrightarrow f=\langle f_\alpha\mid \alpha\in S\rangle$ satisfying for all $\alpha\in S$: $f_\alpha$ is a 2-valued function; $\text{dom}(f_\alpha)$ is a … Continue reading
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
