### 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

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

# Tag Archives: stationary reflection

## 2017 Workshop in Set Theory, Oberwolfach

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 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