### Archives

### Recent blog posts

- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Generalizations of Martin’s Axiom and the well-met condition January 11, 2015
- Many diamonds from just one January 6, 2015
- Happy new jewish year! September 24, 2014
- Square principles April 19, 2014
- Partitioning the club guessing January 22, 2014

### Keywords

Kurepa Hypothesis Whitehead Problem Nonspecial tree Coherent tree Diamond Almost countably chromatic Erdos Cardinal Large Cardinals PFA Almost-disjoint famiy Generalized Clubs Small forcing Erdos-Hajnal graphs Jonsson cardinal Constructible Universe Cardinal function Prikry-type forcing PFA(S)[S] Shelah's Strong Hypothesis free Boolean algebra projective Boolean algebra Rado's conjecture Chang's conjecture coloring number sap Souslin Tree Hedetniemi's conjecture incompactness Fast club tensor product graph 05A17 Foundations square principles polarized partition relation middle diamond Forcing Singular Density Non-saturation Dushnik-Miller Fodor-type reflection Hereditarily Lindelöf space stationary reflection Knaster Successor of Singular Cardinal Universal Sequences Ascent Path Chromatic number Slim tree Successor of Regular Cardinal Poset approachability ideal Almost Souslin Prevalent singular cardinals Square-Brackets Partition Relations Fat stationary set Mandelbrot set stationary hitting Singular cardinals combinatorics reflection principles Absoluteness Minimal Walks Parameterized proxy principle 20M14 diamond star square weak square Ostaszewski square 05D10 P-Ideal Dichotomy L-space Postprocessing function b-scale Reduced Power Hindman's Theorem OCA HOD Partition Relations Uniformization Club Guessing weak diamond Cardinal Invariants Weakly compact cardinal Aronszajn tree Distributive tree Rock n' Roll very good scale Sakurai's Bell inequality Axiom R Forcing Axioms 11P99 Stevo Todorcevic Rainbow sets Singular coﬁnality Antichain S-Space Cohen real ccc Commutative cancellative semigroups Selective Ultrafilter Martin's Axiom Microscopic Approach xbox Uniformly coherent

# Tag Archives: stationary reflection

## MFO workshop in Set Theory, February 2017

I gave an invited talk at the Set Theory workshop in Obwerwolfach, February 2017. Talk Title: Coloring vs. Chromatic. Abstract: In a joint work with Chris Lambie-Hanson, we study the interaction between compactness for the chromatic number (of graphs) and … Continue reading

Posted in Invited Talks
Tagged Chromatic number, coloring number, incompactness, stationary reflection
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## The eightfold way

Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing … Continue reading

## Reflection on the coloring and chromatic numbers

Joint work with Chris Lambie-Hanson. Abstract. We prove that reflection of the coloring number of graphs is consistent with non-reflection of the chromatic number. Moreover, it is proved that incompactness for the chromatic number of graphs (with arbitrarily large gaps) … Continue reading

## The reflection principle $R_2$

A few years ago, in this paper, I introduced the following reflection principle: Definition. $R_2(\theta,\kappa)$ asserts that for every function $f:E^\theta_{<\kappa}\rightarrow\kappa$, there exists some $j<\kappa$ for which the following set is nonstationary: $$A_j:=\{\delta\in E^\theta_\kappa\mid f^{-1}[j]\cap\delta\text{ is nonstationary}\}.$$ I wrote there … Continue reading

Posted in Blog
Tagged reflection principles, square, stationary reflection, Weakly compact cardinal
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## Young Researchers in Set Theory, March 2011

These are the slides of a talk I gave at the Young Researchers in Set Theory 2011 meeting (Königswinter, 21–25 March 2011). Talk Title: Around Jensen’s square principle Abstract: Jensen‘s square principle for a cardinal $\lambda$ asserts the existence of a particular ladder … Continue reading

## Openly generated Boolean algebras and the Fodor-type reflection principle

Joint work with Sakaé Fuchino. Abstract: We prove that the Fodor-type Reflection Principle (FRP) is equivalent to the assertion that any Boolean algebra is openly generated if and only if it is $\aleph _2$-projective. Previously it was known that this … Continue reading

## The failure of diamond on a reflecting stationary set

Joint work with Moti Gitik. Abstract: It is shown that the failure of $\diamondsuit_S$, for a subset $S\subseteq\aleph_{\omega+1}$ that reflects stationarily often, is consistent with GCH and $\text{AP}_{\aleph_\omega}$, relatively to the existence of a supercompact cardinal. This should be comapred with … Continue reading

## A relative of the approachability ideal, diamond and non-saturation

Abstract: Let $\lambda$ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that $\square^*_\lambda$ together with $2^\lambda=\lambda^+$ implies $\diamondsuit_S$ for every $S\subseteq\lambda^+$ that reflects stationarily often. In this paper, for a subset $S\subset\lambda^+$, a normal subideal of … Continue reading