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### 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

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