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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

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

# 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
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## 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
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## 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