### Archives

### 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

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

# 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
Leave a comment

## 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
Leave a comment

## 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