### Archives

### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# 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