Partitioning a cardinal into fat stationary sets, part 1

A subset F of a regular uncountable cardinal kappa is said to be fat iff for every club C in 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 w1 is fat. In particular, w1 may be partitioned into w1 many pairwise disjoint fat sets.

In this talk, I shall prove that square(kappa) give rise to a partition of kappa into kappa many pairwise disjoint fat sets. In particular, the following are equiconsistent:

1. w2 cannot be partitioned into w2 many pairwise disjoint fat sets;
2. w2 cannot be partitioned into two disjoint fat sets;
3. there exists a weakly compact cardinal.