### 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

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

# 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