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### 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

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

# Tag Archives: Shelah’s Strong Hypothesis

## Logic in Hungary, August 2005

These are the slides of a contributed talk given at the Logic in Hungary 2005 meeting (Budapest, 5–11 August 2005). Talk Title: On the consistency strength of the Milner-Sauer Conjecture Abstract: In their paper from 1981, after learning about Pouzet‘s theorem that any … Continue reading

Posted in Contributed Talks
Tagged Antichain, Shelah's Strong Hypothesis, Singular coﬁnality
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## A topological reflection principle equivalent to Shelah’s strong hypothesis

Abstract: We notice that Shelah’s Strong Hypothesis (SSH) is equivalent to the following reflection principle: Suppose $\mathbb X$ is an (infinite) first-countable space whose density is a regular cardinal, $\kappa$. If every separable subspace of $\mathbb X$ is of cardinality at most … Continue reading

Posted in Compactness, Publications, Topology
Tagged 03E04, 03E65, 54G15, Shelah's Strong Hypothesis
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## Openly generated Boolean algebras and the Fodor-type reflection principle

Joint work with Sakaé Fuchino. Abstract: We prove that the Fodor-type Reflection Principle (FRP) is equivalent to the assertion that any Boolean algebra is openly generated if and only if it is $\aleph _2$-projective. Previously it was known that this … Continue reading

## The failure of diamond on a reflecting stationary set

Joint work with Moti Gitik. Abstract: It is shown that the failure of $\diamondsuit_S$, for a subset $S\subseteq\aleph_{\omega+1}$ that reflects stationarily often, is consistent with GCH and $\text{AP}_{\aleph_\omega}$, relatively to the existence of a supercompact cardinal. This should be comapred with … Continue reading

## On the consistency strength of the Milner-Sauer conjecture

Abstract: In their paper from 1981, Milner and Sauer conjectured that for any poset $\mathbb P$, if $\text{cf}(\mathbb P)$ is a singular cardinal $\lambda$, then $\mathbb P$ must contain an antichain of size $\text{cf}(\lambda)$. The conjecture is consistent and known … Continue reading