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### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# Tag Archives: sap

## Jensen’s diamond principle and its relatives

This is chapter 6 in the book Set Theory and Its Applications (ISBN: 0821848127). Abstract: We survey some recent results on the validity of Jensen’s diamond principle at successor cardinals. We also discuss weakening of this principle such as club … 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