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### Recent blog posts

- Square principles April 19, 2014
- Partitioning the club guessing January 22, 2014
- Walk on countable ordinals: the characteristics December 1, 2013
- Polychromatic colorings November 26, 2013
- Universal binary sequences November 14, 2013
- Syndetic colorings with applications to S and L October 26, 2013
- Open coloring and the cardinal invariant $\mathfrak b$ October 8, 2013
- Gabriel Belachsan (14/5/1976 – 20/8/2013) August 20, 2013

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S-Space Prevalent singular cardinals projective Boolean algebra Constructible Universe Square-Brackets Partition Relations Souslin Tree PFA ccc Uniformization Knaster Foundations Cardinal Invariants tensor product graph stationary hitting very good scale stationary reflection Dushnik-Miller Martin's Axiom Universal Sequences PFA(S)[S] Rock n' Roll OCA Aronszajn tree Cardinal function Prikry-type forcing Almost-disjoint famiy Partition Relations Successor of Regular Cardinal Almost countably chromatic Whitehead Problem Diamond Axiom R Hedetniemi's conjecture Rado's conjecture sap Poset incompactness polarized partition relation Erdos Cardinal Hereditarily Lindelöf space Shelah's Strong Hypothesis Singular Density Large Cardinals Singular Cofinality Cohen real Sakurai's Bell inequality b-scale Ostaszewski square weak diamond Erdos-Hajnal graphs P-Ideal Dichotomy Non-saturation Kurepa Hypothesis square Absoluteness Forcing Axioms Weakly compact cardinal Minimal Walks Generalized Clubs L-space Chromatic number weak square free Boolean algebra reflection principles diamond star Small forcing middle diamond Mandelbrot set Singular cardinals combinatorics Rainbow sets Forcing approachability ideal Antichain Club Guessing Successor of Singular Cardinal

# Tag Archives: 03E02

## Rectangular square-bracket operation for successor of regular cardinals

Joint work with Stevo Todorcevic. Extended Abstract: Consider the coloring statement $\lambda^+\nrightarrow[\lambda^+;\lambda^+]^2_{\lambda^+}$ for a given regular cardinal $\lambda$: In 1990, Shelah proved the above for $\lambda>2^{\aleph_0}$; In 1991, Shelah proved the above for $\lambda>\aleph_1$; In 1997, Shelah proved the above … 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