Almost disjoint families in natural models of $AD^+$

Nam Trang (UNT, Denton)

Jun 28. 2024, 15:05 — 15:50

For each cardinal κ, let B(κ) be the ideal of bounded subsets of κ and $P_κ(κ)$ be the ideal of subsets of κ of cardinality less than κ. Assuming $AD^+$, for all $κ < Θ$, there are no maximal B(κ) almost disjoint families A such that $¬(|A| < cof(κ))$. For all $κ < Θ, if cof(κ) > ω$, then there are no maximal $P_κ(κ$ almost disjoint families A so that $¬(|A| < cof(κ))$.

This is joint work with W. Chan and S. Jackson. Our work is inspired by work of Schrittesser and Törnquist, and of Neeman and Norwood that showed there are no maximal almost disjoint families on ω under $AD^+$. 

Further Information
Venue:
ESI Boltzmann Lecture Hall
Recordings:
Recording
Associated Event:
Determinacy, Inner Models and Forcing Axioms (Workshop)
Organizer(s):
Sandra Müller (TU Vienna)
Grigor Sargsyan (Polish Academy of Science, Warsaw)
Ralf Schindler (WWU Münster)
John Steel (UC, Berkeley)