### 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

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

# 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