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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

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

# 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