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### Recent blog posts

- 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

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

# 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