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Forcing Axioms Sigma-Prikry very good scale Rock n' Roll Almost-disjoint family Axiom R Partition relations for trees Postprocessing function Singular cofinality perfectly normal Generalized Clubs xbox stationary reflection Distributive tree Subtle cardinal polarized partition relation club_AD Kurepa Hypothesis square principles Generalized descriptive set theory Commutative cancellative semigroups Strongly Luzin set Filter reflection Aronszajn tree Almost countably chromatic Erdos Cardinal OCA Almost Souslin Precaliber Prevalent singular cardinals Jonsson cardinal middle diamond Entangled linear order Selective Ultrafilter Rainbow sets Diamond for trees Ostaszewski square Forcing with side conditions sap Non-saturation Diamond-sharp Poset Rado's conjecture approachability ideal SNR Fast club Chang's conjecture PFA(S)[S] positive partition relation free Souslin tree Singular cardinals combinatorics indecomposable filter square Souslin Tree b-scale Respecting tree nonmeager set Sakurai's Bell inequality Was Ulam right? Minimal Walks Singular Density PFA unbounded function Ascending path Large Cardinals tensor product graph Ineffable cardinal Partition Relations stationary hitting Club Guessing Local Club Condensation. Diamond Ulam matrix Shelah's Strong Hypothesis Analytic sets Monotonically far Forcing Iterated forcing higher Baire space Intersection model Uniformly homogeneous weak diamond Knaster and friends reflection principles Prikry-type forcing Fodor-type reflection HOD Constructible Universe specializable Souslin tree full tree Small forcing Ramsey theory over partitions coloring number Dushnik-Miller O-space Vanishing levels Nonspecial tree Lipschitz reduction Universal Sequences Foundations free Boolean algebra Weakly compact cardinal C-sequence AIM forcing Martin's Axiom ZFC construction Well-behaved magma weak square P-Ideal Dichotomy GMA Amenable C-sequence Reduced Power Strongly compact cardinal 54G20 Knaster Subtle tree property Hereditarily Lindelöf space Uniformization Cardinal function Antichain Open Access regressive Souslin tree L-space Luzin set Chromatic number Successor of Singular Cardinal Mandelbrot set Subadditive Cardinal Invariants weak Kurepa tree Successor of Regular Cardinal Uniformly coherent Absoluteness Microscopic Approach Erdos-Hajnal graphs Cohen real countably metacompact transformations Sierpinski's onto mapping principle incompactness strongly bounded groups stick Greatly Mahlo Subnormal ideal Parameterized proxy principle Reflecting stationary set Commutative projection system super-Souslin tree Hedetniemi's conjecture Coherent tree Fat stationary set Strong coloring Countryman line Ascent Path Closed coloring Dowker space Slim tree projective Boolean algebra Whitehead Problem Hindman's Theorem ccc S-Space Interval topology on trees diamond star Square-Brackets Partition Relations
Author Archives: Assaf Rinot
Set Theory Programme on Large Cardinals and Forcing, September 2013
I gave an invited talk at the Large Cardinals and Forcing meeting, Erwin Schrödinger International Institute for Mathematical Physics, Vienna, September 23–27, 2013. Talk Title: Hedetniemi’s conjecture for uncountable graphs Abstract: It is proved that in Godel’s constructible universe, for … Continue reading
Posted in Invited Talks
Tagged Almost countably chromatic, Chromatic number, Hedetniemi's conjecture
1 Comment
PFA and the tree property at $\aleph_2$
Recall that a poset $\langle T,\le\rangle$ is said to be a $\lambda^+$-Aronszajn tree, if it isomorphic to a poset $(\mathcal T,\subseteq)$ of the form: $\emptyset\in \mathcal T\subseteq{}^{<\lambda^+}\lambda$; Write $\mathcal T_\alpha:=\{\sigma\in\mathcal T\mid \text{dom}(\sigma)=\alpha\}$; for all $\alpha<\lambda^+$, $\mathcal T_\alpha$ has size $\le\lambda$, … Continue reading
A Kurepa tree from diamond-plus
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
Posted in Blog, Expository
Tagged diamond star, Kurepa Hypothesis
Comments Off on A Kurepa tree from diamond-plus
Chromatic numbers of graphs – large gaps
Abstract. We say that a graph $G$ is $(\aleph_0,\kappa)$-chromatic if $\text{Chr}(G)=\kappa$, while $\text{Chr}(G’)\le\aleph_0$ for any subgraph $G’$ of $G$ of size $<|G|$. The main result of this paper reads as follows. If $\square_\lambda+\text{CH}_\lambda$ holds for a given uncountable cardinal $\lambda$, … Continue reading
Posted in Compactness, Infinite Graphs, Publications
Tagged 03E35, 05C15, 05C63, Almost countably chromatic, Chromatic number, incompactness, Ostaszewski square
6 Comments
The S-space problem, and the cardinal invariant $\mathfrak b$
Recall that an S-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In a previous post, we showed that such a space exists after adding a Cohen real. Here, we shall construct one from an arithmetic … Continue reading
An $S$-space from a Cohen real
Recall that an $S$-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. In this post, we shall establish the consistency of the existence of such a space. Theorem (Roitman, 1979). Let $\mathbb C=({}^{<\omega}\omega,\subseteq)$ be the notion of … Continue reading
Forcing with a Souslin tree makes $\mathfrak p=\omega_1$
I was meaning to include a proof of Farah’s lemma in my previous post, but then I realized that the slick proof assumes some background which may worth spelling out, first. Therefore, I am dedicating a short post for a … Continue reading
The S-space problem, and the cardinal invariant $\mathfrak p$
Recall that an $S$-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. Do they exist? Consistently, yes. However, Szentmiklóssy proved that compact $S$-spaces do not exist, assuming Martin’s Axiom. Pushing this further, Todorcevic later proved that … Continue reading
Posted in Blog, Expository, Open Problems
Tagged Cardinal Invariants, Hereditarily Lindelöf space, P-Ideal Dichotomy, PFA(S)[S], S-Space
4 Comments
Jones’ theorem on the cardinal invariant $\mathfrak p$
This post continues the study of the cardinal invariant $\mathfrak p$. We refer the reader to a previous post for all the needed background. For ordinals $\alpha,\alpha_0,\alpha_1,\beta,\beta_0,\beta_1$, the polarized partition relation $$\left(\begin{array}{c}\alpha\\\beta\end{array}\right)\rightarrow\left(\begin{array}{cc}\alpha_0&\alpha_1\\\beta_0&\beta_1\end{array}\right)$$ asserts that for every coloring $f:\alpha\times\beta\rightarrow 2$, (at least) … Continue reading
Erdős 100
The influential mathematician Paul Erdős was born 100 years ago, 26 March 1913, in Budapest. One evidence of his impact on mathematics is reflected in the particular list of invited speakers for the upcoming conference in his honor. Erdős is also … Continue reading
