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### Recent blog posts

- Genearlizations 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
- Walk on countable ordinals: the characteristics December 1, 2013
- Polychromatic colorings November 26, 2013
- Universal binary sequences November 14, 2013

### Keywords

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

# 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