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### 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

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

# 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 Publications, Squares and Diamonds
Tagged 03E05, 03E45, Almost Souslin, diamond star, Kurepa Hypothesis, Minimal Walks, square, xbox
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## 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