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- Prikry forcing may add a Souslin tree June 12, 2016
- 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

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Universal Sequences Partition Relations Minimal Walks incompactness Martin's Axiom Hereditarily Lindelöf space Antichain square Hedetniemi's conjecture OCA PFA(S)[S] S-Space Absoluteness Dushnik-Miller Ostaszewski square Prikry-type forcing Axiom R polarized partition relation stationary reflection Forcing Axioms ccc Large Cardinals Whitehead Problem Cardinal function Erdos Cardinal Souslin Tree Fast club Chang's conjecture free Boolean algebra Kurepa Hypothesis very good scale approachability ideal Shelah's Strong Hypothesis Diamond Sakurai's Bell inequality Weakly compact cardinal Fat stationary set Slim tree Non-saturation Coherent tree Singular Cofinality 20M14 11P99 middle diamond tensor product graph Reduced Power Rado's conjecture Rock n' Roll Constructible Universe Chromatic number Stevo Todorcevic projective Boolean algebra Cardinal Invariants Club Guessing PFA Successor of Regular Cardinal stationary hitting Hindman's Theorem Uniformization Forcing Prevalent singular cardinals 05A17 Jonsson cardinal Ascent Path Almost countably chromatic Commutative cancellative semigroups Microscopic Approach Singular Density Almost Souslin Generalized Clubs Rainbow sets reflection principles HOD Singular cardinals combinatorics Successor of Singular Cardinal Cohen real Square-Brackets Partition Relations Aronszajn tree Knaster weak square diamond star P-Ideal Dichotomy Small forcing 05D10 Foundations Singular coﬁnality Selective Ultrafilter Erdos-Hajnal graphs Mandelbrot set Poset Parameterized proxy principle weak diamond b-scale sap Almost-disjoint famiy L-space

# Tag Archives: 03E02

## Strong failures of higher analogs of Hindman’s Theorem

Joint work with David J. Fernández Bretón. Abstract. We show that various analogs of Hindman’s Theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets: Theorem 1. There exists a colouring $c:\mathbb R\rightarrow\mathbb Q$, such that … Continue reading

Posted in Partition Relations, Preprints
Tagged 03E02, 03E35, 03E75, 05A17, 05D10, 11P99, 20M14, Chang's conjecture, Commutative cancellative semigroups, Erdos Cardinal, Hindman's Theorem, Jonsson cardinal, Kurepa Hypothesis, Square-Brackets Partition Relations, Weakly compact cardinal
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## 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