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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

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

# 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