### 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

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

# 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