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### Recent blog posts

- A strong form of König’s lemma October 21, 2017
- 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

### Keywords

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

# 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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