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