### 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

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

# 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