### 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

xbox PFA(S)[S] coloring number Singular Cofinality Erdos-Hajnal graphs Prikry-type forcing Jonsson cardinal 05A17 Uniformization Microscopic Approach reflection principles Cardinal function Singular Density Forcing Chang's conjecture b-scale Successor of Regular Cardinal Almost-disjoint famiy Universal Sequences Poset Rado's conjecture HOD incompactness Axiom R Club Guessing Erdos Cardinal S-Space Successor of Singular Cardinal 11P99 Stevo Todorcevic Prevalent singular cardinals stationary hitting Selective Ultrafilter Non-saturation Square-Brackets Partition Relations projective Boolean algebra Ascent Path Rock n' Roll Chromatic number Reduced Power Commutative cancellative semigroups diamond star Fast club Diamond Hindman's Theorem tensor product graph Aronszajn tree Mandelbrot set 05D10 Almost Souslin Forcing Axioms Partition Relations Kurepa Hypothesis Slim tree Dushnik-Miller Shelah's Strong Hypothesis Generalized Clubs stationary reflection P-Ideal Dichotomy free Boolean algebra Parameterized proxy principle ccc Cardinal Invariants square very good scale Minimal Walks Foundations square principles Singular cardinals combinatorics L-space sap Antichain Sakurai's Bell inequality Weakly compact cardinal Small forcing Whitehead Problem Fat stationary set Almost countably chromatic 20M14 Large Cardinals Constructible Universe Souslin Tree middle diamond OCA polarized partition relation weak diamond Absoluteness Hedetniemi's conjecture Martin's Axiom Rainbow sets PFA Knaster Cohen real approachability ideal Ostaszewski square Coherent tree Fodor-type reflection Singular coﬁnality weak square Hereditarily Lindelöf space

# Tag Archives: Universal Sequences

## 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

## Syndetic colorings with applications to S and L

Notation. Write $\mathcal Q(A):=\{ a\subseteq A\mid a\text{ is finite}, a\neq\emptyset\}$. Definition. An L-space is a regular hereditarily Lindelöf topological space which is not hereditarily separable. Definition. We say that a coloring $c:[\omega_1]^2\rightarrow\omega$ is L-syndetic if the following holds. For every uncountable … Continue reading