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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

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

# 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