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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

### Keywords

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