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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

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

# 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