### 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

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