### Archives

### Recent blog posts

- 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
- 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

### Keywords

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

## 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

## The failure of diamond on a reflecting stationary set

Joint work with Moti Gitik. Abstract: It is shown that the failure of $\diamondsuit_S$, for a subset $S\subseteq\aleph_{\omega+1}$ that reflects stationarily often, is consistent with GCH and $\text{AP}_{\aleph_\omega}$, relatively to the existence of a supercompact cardinal. This should be comapred with … 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