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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

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

# 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