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