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