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### Recent blog posts

- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Genearlizations of Martin’s Axiom and the well-met condition January 11, 2015
- Many diamonds from just one January 6, 2015
- 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

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

# 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