### 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

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

# 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