### 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

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

# Tag Archives: Erdos Cardinal

## 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

## A large cardinal in the constructible universe

In this post, we shall provide a proof of Silver’s theorem that the Erdos caridnal $\kappa(\omega)$ relativizes to Godel’s constructible universe. First, recall some definitions. Given a function $f:[\kappa]^{<\omega}\rightarrow \mu$, we say that $I\subseteq\kappa$ is a set of indiscernibles for … Continue reading