### 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
- 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

### Keywords

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

# 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