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### Recent blog posts

- 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
- Gabriel Belachsan (14/5/1976 – 20/8/2013) August 20, 2013

### Keywords

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