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- 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

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

# 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