### Archives

### 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

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