Partitioning Subsets of Stable Models

This paper discusses two combinatorial problems in stability theory. First we prove a partition result for subsets of stable models: for any A and B, we can partition A into |B | ) pieces, 〈Ai | i < |B | 〉, such that for each Ai there is a Bi ⊆ B where |Bi| < κ(T ) and Ai |̂ Bi B. Second, if A and B are as above and |A | > |B |, then we try to find A′ ⊂ A and B′ ⊂ B such that |A′| is as large as possible, |B′| is as small as possible, and A′ |̂ B′ B. We prove some positive results in this direction, and we then discuss the optimality of these results under ZFC + GCH.