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- A strong form of König’s lemma October 21, 2017
- 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

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

# 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