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Singular cofinality Cohen real Commutative projection system PFA(S)[S] Chromatic number Partition relations for trees Almost countably chromatic Large Cardinals Was Ulam right? weak diamond Knaster Uniformization Almost-disjoint family Subnormal ideal full tree xbox Ramsey theory over partitions Subtle tree property Coherent tree AIM forcing Filter reflection Commutative cancellative semigroups Diamond for trees Cardinal Invariants square Precaliber Dushnik-Miller Singular cardinals combinatorics Successor of Singular Cardinal Whitehead Problem Fast club L-space Microscopic Approach C-sequence Local Club Condensation. Parameterized proxy principle Axiom R countably metacompact Mandelbrot set Weakly compact cardinal stick free Boolean algebra Universal Sequences reflection principles Strong coloring weak square Erdos Cardinal b-scale projective Boolean algebra Hedetniemi's conjecture PFA Closed coloring coloring number super-Souslin tree Square-Brackets Partition Relations Strongly compact cardinal Nonspecial tree sap Selective Ultrafilter Luzin set Antichain Ulam matrix Sierpinski's onto mapping principle middle diamond Respecting tree Sakurai's Bell inequality Singular Density Aronszajn tree Fodor-type reflection Erdos-Hajnal graphs Almost Souslin Strongly Luzin set Uniformly coherent Subadditive transformations stationary hitting Foundations Diamond-sharp Minimal Walks Prevalent singular cardinals Interval topology on trees Ascending path Slim tree very good scale Dowker space Prikry-type forcing OCA Ineffable cardinal strongly bounded groups nonmeager set Forcing with side conditions Ostaszewski square P-Ideal Dichotomy Postprocessing function Forcing Axioms Successor of Regular Cardinal Partition Relations Small forcing Absoluteness Uniformly homogeneous Vanishing levels Poset Kurepa Hypothesis O-space Greatly Mahlo Souslin Tree tensor product graph Subtle cardinal regressive Souslin tree indecomposable filter 54G20 approachability ideal Open Access Jonsson cardinal Iterated forcing Rock n' Roll Shelah's Strong Hypothesis square principles Rado's conjecture stationary reflection Forcing incompactness diamond star club_AD Diamond S-Space Reflecting stationary set Rainbow sets Knaster and friends Chang's conjecture GMA Club Guessing Martin's Axiom higher Baire space Well-behaved magma Entangled linear order Fat stationary set Non-saturation Lipschitz reduction SNR ccc Intersection model Distributive tree Monotonically far unbounded function Hereditarily Lindelöf space Analytic sets positive partition relation perfectly normal Sigma-Prikry Ascent Path Reduced Power Hindman's Theorem ZFC construction weak Kurepa tree HOD specializable Souslin tree Generalized Clubs Constructible Universe Cardinal function polarized partition relation Amenable C-sequence Countryman line free Souslin tree Generalized descriptive set theory
Category Archives: Blog
Polychromatic colorings
These are lectures notes of two talks Dani Livne gave in our Infinite Combinatorics seminar. I did not take notes in real-time, hence, all possible mistakes here are due to myself. Recall that a function $f:A\rightarrow B$ is said to … Continue reading
Universal binary sequences
Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Suppose for the moment that we are given a fixed sequence $\langle f_\alpha:\omega\rightarrow2\mid \alpha\in a\rangle$, indexed by some set $a$ of ordinals. Then, for every function $h:a\rightarrow\omega$ and $i<\omega$, we … Continue reading
Syndetic colorings with applications to S and L
Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Definition. An L-space is a regular hereditarily Lindelöf topological space which is not hereditarily separable. Definition. We say that a coloring $c:[\omega_1]^2\rightarrow\omega$ is L-syndetic if the following holds. For every uncountable … Continue reading
Open coloring and the cardinal invariant $\mathfrak b$
Nik Weaver asked for a direct proof of the fact that Todorcevic’s axiom implies the failure of CH fails. Here goes. Notation. For a set $X$, we write $[X]^2$ for the set of unordered pairs $\{ \{x,x’\}\mid x,x’\in X, x\neq … Continue reading
Gabriel Belachsan (14/5/1976 – 20/8/2013)
רק כשעיני סגורות, עולם נגלה לפני
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
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Tagged diamond star, Kurepa Hypothesis
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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
