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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

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

# 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