### Archives

### Recent blog posts

- Square principles April 19, 2014
- Partitioning the club guessing January 22, 2014
- Walk on countable ordinals: the characteristics December 1, 2013
- Polychromatic colorings November 26, 2013
- Universal binary sequences November 14, 2013
- Syndetic colorings with applications to S and L October 26, 2013
- Open coloring and the cardinal invariant $\mathfrak b$ October 8, 2013
- Gabriel Belachsan (14/5/1976 – 20/8/2013) August 20, 2013

### Keywords

Cardinal Invariants Kurepa Hypothesis Sakurai's Bell inequality Almost-disjoint famiy PFA(S)[S] Singular Density square sap Rainbow sets Large Cardinals very good scale Forcing middle diamond diamond star weak square Successor of Regular Cardinal Foundations Forcing Axioms Dushnik-Miller projective Boolean algebra Rado's conjecture S-Space Universal Sequences Cohen real Erdos Cardinal approachability ideal Absoluteness Erdos-Hajnal graphs Small forcing polarized partition relation Singular Cofinality weak diamond Souslin Tree Knaster Hereditarily Lindelöf space Whitehead Problem Rock n' Roll free Boolean algebra Prevalent singular cardinals L-space Mandelbrot set incompactness Generalized Clubs Constructible Universe Hedetniemi's conjecture tensor product graph Poset P-Ideal Dichotomy Singular cardinals combinatorics Uniformization Square-Brackets Partition Relations Diamond Cardinal function OCA Axiom R b-scale Minimal Walks Antichain stationary reflection Successor of Singular Cardinal PFA Aronszajn tree Ostaszewski square Martin's Axiom Almost countably chromatic Chromatic number stationary hitting reflection principles Shelah's Strong Hypothesis Partition Relations Non-saturation Club Guessing Prikry-type forcing### Name Dropping

Ace Billet Alan Mekler Albin L. Jones Alex Primavesi Alfred Tarski András Hajnal Benoit Mandelbrot Boban Velickovic Chen Meiri Chris Hadfield Ernest Schimmerling Fred Glavin Gabriel Belachsan Hiroshi Sakai Ilijas Farah Itay Neeman Jack Silver Jim Baumgartner John Krueger Judy Roitman Keith Devlin Menachem Magidor Mirna Dzamonja Moti Gitik Murray Bell Paul Erdős Paul Larson Richard Laver Ronald Jensen Saharon Shelah Sakaé Fuchino Stevo Todorcevic Teruyuki Yorioka Wacław Sierpiński

# Category Archives: Blog

## Square principles

Since the birth of Jensen’s original Square principle, many variations of the principle were introduced and intensively studied. Asaf Karagila suggested me today to put some order into all of these principles. Here is a trial. Definition. A square principle … Continue reading

## Partitioning the club guessing

In a recent paper, I am making use of the following fact. Theorem (Shelah, 1997). Suppose that $\kappa$ is an accessible cardinal (i.e., there exists a cardinal $\theta<\kappa$ such that $2^\theta\ge\kappa)$. Then there exists a sequence $\langle g_\delta:C_\delta\rightarrow\omega\mid \delta\in E^{\kappa^+}_\kappa\rangle$ … Continue reading

## Walk on countable ordinals: the characteristics

In this post, we shall present a few aspects of the method of walk on ordinals (focusing on countable ordinals), record its characteristics, and verify some of their properties. All definitions and results in this post are due to Todorcevic. … Continue reading

## 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

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Tagged L-space, S-Space, Universal Sequences
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## 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