### 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

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

# 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