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### Recent blog posts

- Partitioning the club guessing January 22, 2014
- Walk on countable ordinals: the characteristics December 1, 2013
- Polychromatic colorings November 26, 2013
- Universal binary sequences November 14, 2013
- Syndetic colorings with applications to S and L October 26, 2013
- Open coloring and the cardinal invariant $\mathfrak b$ October 8, 2013
- Gabriel Belachsan (14/5/1976 – 20/8/2013) August 20, 2013
- PFA and the tree property at $\aleph_2$ June 9, 2013

### Keywords

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

- Luzin sets and generalizations
- Nonuniversal colorings in ZFC
- Large Sets
- Infinite-dimensional Jonsson algebras
- Strong colorings without nontrivial polychromatic sets
- Infinite-dimensional polychromatic colorings
- Polychromatic colorings of the first uncountable cardinal
- From colorings to topology
- From topology to colorings
- Anti-Ramsey colorings of the rational numbers, part 2

# Tag Archives: diamond star

## 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

Posted in Open Problems, Publications
Tagged 03E05, 03E35, 03E50, approachability ideal, Club Guessing, Diamond, diamond star, Non-saturation, sap, Souslin Tree, square, stationary hitting, Uniformization, Whitehead Problem
2 Comments

## 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