### 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

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

# Category Archives: Singular Cardinals Combinatorics

## Ordinal definable subsets of singular cardinals

Joint work with James Cummings, Sy-David Friedman, Menachem Magidor, and Dima Sinapova. Abstract. A remarkable result by Shelah states that if $\kappa$ is a singular strong limit cardinal of uncountable cofinality then there is a subset $x$ of $\kappa$ such … Continue reading

Posted in Publications, Singular Cardinals Combinatorics
Tagged HOD, Singular cardinals combinatorics
2 Comments

## Aspects of singular cofinality

Abstract. We study properties of closure operators of singular cofinality, and introduce several ZFC sufficient and equivalent conditions for the existence of antichain sequences in posets of singular cofinality. We also notice that the Proper Forcing Axiom implies the Milner-Sauer … 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