### Archives

### 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

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

# 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
1 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