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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

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

# Tag Archives: Shelah’s Strong Hypothesis

## Logic in Hungary 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