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