### 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

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

# 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