### 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

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