### 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

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

# Tag Archives: Almost-disjoint famiy

## Universal binary sequences

Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Suppose for the moment that we are given a fixed sequence $\langle f_\alpha:\omega\rightarrow2\mid \alpha\in a\rangle$, indexed by some set $a$ of ordinals. Then, for every function $h:a\rightarrow\omega$ and $i<\omega$, we … Continue reading

## The Engelking-Karlowicz theorem, and a useful corollary

Theorem (Engelking-Karlowicz, 1965). For cardinals $\kappa\le\lambda\le\mu\le 2^\lambda$, the following are equivalent: $\lambda^{<\kappa}=\lambda$; there exists a collection of functions, $\langle f_i:\mu\rightarrow\lambda\mid i<\lambda\rangle$, such that for every $X\in[\mu]^{<\kappa}$ and every function $f:X\rightarrow\lambda$, there exists some $i<\lambda$ with $f\subseteq f_i$. Proof. (2)$\Rightarrow$(1) Suppose … Continue reading