### 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

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

# 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