### 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

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

# 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