### Archives

### Recent blog posts

- Prolific Souslin trees March 17, 2016
- Genearlizations 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
- Walk on countable ordinals: the characteristics December 1, 2013
- Polychromatic colorings November 26, 2013

### Keywords

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

# Tag Archives: diamond star

## Square with built-in diamond-plus

Joint work with Ralf Schindler. Abstract. We formulate combinatorial principles that combine the square principle with various strong forms of diamond, and prove that the strongest amongst them holds in $L$ for every infinite cardinal. As an application, we prove that … Continue reading

Posted in Preprints, Squares and Diamonds
Tagged 03E05, 03E45, Almost Souslin, diamond star, Kurepa Hypothesis, Minimal Walks, square
Leave a comment

## 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

## 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

## 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