### 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

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

# 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