### Archives

### 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

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

# 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