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