### 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

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

# Tag Archives: P-Ideal Dichotomy

## The S-space problem, and the cardinal invariant $\mathfrak p$

Recall that an $S$-space is a regular hereditarily separable topological space which is not hereditarily Lindelöf. Do they exist? Consistently, yes. However, Szentmiklóssy proved that compact $S$-spaces do not exist, assuming Martin’s Axiom. Pushing this further, Todorcevic later proved that … Continue reading

Posted in Blog, Expository, Open Problems
Tagged Hereditarily Lindelöf space, P-Ideal Dichotomy, PFA(S)[S], S-Space
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## The P-Ideal Dichotomy and the Souslin Hypothesis

John Krueger is visiting Toronto these days, and in a conversation today, we asked ourselves how do one prove the Abraham-Todorcevic theorem that PID implies SH. Namely, that the next statement implies that there are no Souslin trees: Definition. The … Continue reading

## Dushnik-Miller for regular cardinals (part 3)

Here is what we already know about the Dushnik-Miller theorem in the case of $\omega_1$ (given our earlier posts on the subject): $\omega_1\rightarrow(\omega_1,\omega+1)^2$ holds in ZFC; $\omega_1\rightarrow(\omega_1,\omega+2)^2$ may consistently fail; $\omega_1\rightarrow(\omega_1,\omega_1)^2$ fails in ZFC. In this post, we shall provide … Continue reading