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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

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

# 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