### 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

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

# 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