### 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

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

# 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