### 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

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

# 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