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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

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

# 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
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## 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
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## 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
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