### 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

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

# 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