### 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

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

# 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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