### 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

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

# 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