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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

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

# 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