All about mad families

I will give an overview of the developments in the past 5 years regarding mad families.

 

We'll study families of subsets of the natural numbers, and say that such a family is almost disjoint if any two distinct elements intersect finitely. The Axiom of Choice implies the existence of infinite almost disjoint family which is maximal under inclusion.

 

Mathias proved in the late 1960s that it is consistent with ZF+DC that there are no mad families. He needed a Mahlo cardinal to do this.  In 2014 I showed that the classical Solovay-Lévy model has no infinite mad families, and shortly thereafter, in 2016, Horowitz and Shelah showed that you don't even need an inaccessible to get a model of ZF+DC+no infinite mad families.

 

A wealth of related questions have also been settled, most recently, I have shown with David Schrittesser that "All sets are Ramsey"+"Ramsey uniformization" implies "no infinite mad families ".

 

I'll also discuss open problems. The talk will not assume any prior knowledge of mad families.