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

# 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