### Archives

### Recent blog posts

- 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
- Syndetic colorings with applications to S and L October 26, 2013
- Open coloring and the cardinal invariant $\mathfrak b$ October 8, 2013

### Keywords

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

# 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