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### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# 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