BIU seminar in Set Theory
On 29/12/2016, 10-12, Building 216, Room 201
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:
- $\omega_2$ cannot be partitioned into $\omega_2$ many pairwise disjoint fat sets;
- $\omega_2$ cannot be partitioned into two disjoint fat sets;
- there exists a weakly compact cardinal.