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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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Weakly compact cardinal projective Boolean algebra very good scale Rainbow sets Prevalent singular cardinals sap xbox Fast club 05A17 Commutative cancellative semigroups 11P99 Chromatic number Reduced Power Kurepa Hypothesis Erdos-Hajnal graphs Parameterized proxy principle b-scale Universal Sequences Jonsson cardinal Hindman's Theorem Mandelbrot set Singular cardinals combinatorics Foundations Successor of Singular Cardinal Fodor-type reflection Uniformization diamond star Hereditarily Lindelöf space approachability ideal Souslin Tree Antichain Constructible Universe Cohen real stationary reflection reflection principles Singular coﬁnality Cardinal function Hedetniemi's conjecture Selective Ultrafilter Poset 05D10 Fat stationary set Diamond Martin's Axiom Coherent tree Chang's conjecture Aronszajn tree Uniformly coherent Microscopic Approach Ascent Path Partition Relations Distributive tree weak diamond Prikry-type forcing Minimal Walks Whitehead Problem stationary hitting Almost Souslin Small forcing free Boolean algebra Square-Brackets Partition Relations Nonspecial tree OCA Slim tree incompactness Erdos Cardinal Almost-disjoint famiy Rado's conjecture Cardinal Invariants L-space S-Space Sakurai's Bell inequality Forcing Axioms Ostaszewski square Dushnik-Miller Almost countably chromatic PFA Axiom R Club Guessing Non-saturation Stevo Todorcevic coloring number Large Cardinals Successor of Regular Cardinal Rock n' Roll Forcing ccc HOD 20M14 Generalized Clubs square Postprocessing function weak square square principles P-Ideal Dichotomy middle diamond PFA(S)[S] Knaster polarized partition relation Singular Density Absoluteness Shelah's Strong Hypothesis tensor product graph

# 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