### Archives

### Recent blog posts

- Prikry forcing may add a Souslin tree June 12, 2016
- The reflection principle $R_2$ May 20, 2016
- Prolific Souslin trees March 17, 2016
- Generalizations 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

### Keywords

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

# 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

## 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