### 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

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

# 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