Assaf Rinot: Partitioning a cardinal into fat stationary sets

BIU seminar in Set Theory

On 29/12/2016, 10-12, Building 216, Room 201

  Assaf Rinot

Partitioning a cardinal into fat stationary sets

A subset $F$ of a regular uncountable cardinal $\kappa$ is said to be fat iff for every club $C\subseteq\kappa$, and every ordinal $\alpha<\kappa$, $F\cap C$ contains a closed copy of $\alpha+1$.
By a theorem of H. Friedman from 1974, every stationary subset of $\omega_1$ is fat. In particular, $\omega_1$ may be partitioned into $\omega_1$ many pairwise disjoint fat sets.

In this talk, I shall prove that $\square(\kappa)$ implies that any fat subset of $\kappa$ may be partitioned into $\kappa$ many pairwise disjoint fat sets. In particular, the following are equiconsistent:

  1. $\omega_2$ cannot be partitioned into $\omega_2$ many pairwise disjoint fat sets;
  2. $\omega_2$ cannot be partitioned into two disjoint fat sets;
  3. there exists a weakly compact cardinal.

Leave a Reply

Your email address will not be published. Required fields are marked *

Time limit is exhausted. Please reload CAPTCHA.