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- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Genearlizations 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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20M14 middle diamond ccc Almost countably chromatic Sakurai's Bell inequality S-Space Fodor-type reflection Aronszajn tree Singular Density Ostaszewski square Erdos Cardinal Successor of Regular Cardinal projective Boolean algebra stationary hitting incompactness Cardinal Invariants Prevalent singular cardinals Singular coﬁnality Selective Ultrafilter PFA(S)[S] coloring number Cardinal function 11P99 Diamond Erdos-Hajnal graphs P-Ideal Dichotomy L-space Minimal Walks diamond star Club Guessing Prikry-type forcing Souslin Tree Rock n' Roll Forcing Axioms sap Knaster PFA Fat stationary set Antichain b-scale HOD Martin's Axiom Fast club Hereditarily Lindelöf space OCA Generalized Clubs weak square Forcing polarized partition relation Uniformization weak diamond Chang's conjecture Hedetniemi's conjecture Poset Weakly compact cardinal approachability ideal Jonsson cardinal tensor product graph Singular cardinals combinatorics Mandelbrot set Cohen real Slim tree xbox Parameterized proxy principle Stevo Todorcevic Constructible Universe Hindman's Theorem 05A17 Non-saturation Microscopic Approach Singular Cofinality Rado's conjecture stationary reflection Commutative cancellative semigroups Square-Brackets Partition Relations Large Cardinals Shelah's Strong Hypothesis Dushnik-Miller Small forcing Coherent tree Almost-disjoint famiy Successor of Singular Cardinal reflection principles 05D10 very good scale Universal Sequences square Chromatic number Foundations Axiom R Partition Relations Ascent Path Reduced Power Absoluteness Whitehead Problem Almost Souslin free Boolean algebra Kurepa Hypothesis Rainbow sets

# 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