### 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

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

# Tag Archives: 03E04

## A cofinality-preserving small forcing may introduce a special Aronszajn tree

Extended Abstract: Shelah proved that Cohen forcing introduces a Souslin tree; Jensen proved that a c.c.c. forcing may consistently add a Kurepa tree; Todorcevic proved that a Knaster poset may already force the Kurepa hypothesis; Irrgang introduced a c.c.c. notion … Continue reading

Posted in Publications, Squares and Diamonds
Tagged 03E04, 03E05, 03E35, Aronszajn tree, Small forcing, Successor of Singular Cardinal, weak square
Leave a comment

## On topological spaces of singular density and minimal weight

Abstract: We introduce a weakening of the Generalized Continuum Hypothesis, which we will refer to as the Prevalent Singular cardinals Hypothesis (PSH), and show it implies that every topological space of density and weight $\aleph_{\omega_1}$ is not hereditarily Lindelöf. The assumption … Continue reading

## 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

## 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

## Antichains in partially ordered sets of singular cofinality

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 main result of of this … Continue reading

Posted in Publications, Singular Cardinals Combinatorics
Tagged 03E04, 03E35, 06A07, Antichain, Poset, Singular coﬁnality
Leave a comment