### 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
- Genearlizations 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

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

# 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