### Archives

### 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

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

# Tag Archives: Successor of Singular Cardinal

## Putting a diamond inside the square

Abstract. By a 35-year-old theorem of Shelah, $\square_\lambda+\diamondsuit(\lambda^+)$ does not imply square-with-built-in-diamond_lambda for regular uncountable cardinals $\lambda$. Here, it is proved that $\square_\lambda+\diamondsuit(\lambda^+)$ is equivalent to square-with-built-in-diamond_lambda for every singular cardinal $\lambda$. Downloads: Citation information: A. Rinot, Putting a diamond inside … Continue reading

Posted in Publications, Squares and Diamonds
Tagged 03E05, 03E45, Diamond, square, Successor of Singular Cardinal
1 Comment

## A cofinality-preserving small forcing may introduce a special Aronszajn tree

Extended Abstract: Shelah proved that Cohen forcing introduces a Souslin tree; Jensen proved that a c.c.c. forcing may consistently add a Kurepa tree; Todorcevic proved that a Knaster poset may already force the Kurepa hypothesis; Irrgang introduced a c.c.c. notion … Continue reading

Posted in Publications, Squares and Diamonds
Tagged 03E04, 03E05, 03E35, Aronszajn tree, Small forcing, Successor of Singular Cardinal, weak square
Leave a comment

## 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

## Transforming rectangles into squares, with applications to strong colorings

Abstract: It is proved that every singular cardinal $\lambda$ admits a function $\textbf{rts}:[\lambda^+]^2\rightarrow[\lambda^+]^2$ that transforms rectangles into squares. That is, whenever $A,B$ are cofinal subsets of $\lambda^+$, we have $\textbf{rts}[A\circledast B]\supseteq C\circledast C$, for some cofinal subset $C\subseteq\lambda^+$. As a … Continue reading