### 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

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

# 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