### Archives

### 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

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

# 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