The amalgamation spectrum

We study when classes can have the disjoint amalgamation property for a proper initial segment of cardinals. Theorem A For every natural number k, there is a class K_k defined by a sentence in L_{ω1. ω} that has no models of cardinality greater than ⊃_k+1, but K _k has the disjoint amalgamation property on models of cardinality less than or equal to N_k-3 and has models of cardinality N _k-3. More strongly, we can have disjoint amalgamation up to N _α for α < ω₁, but have a bound on size of models. Theorem B For every countable ordinal α, there is a class K_α defined by a sentence in L_ω1.ω that has no models of cardinality greater than ⊃_ω1, but K does have the disjoint amalgamation property on models of cardinality less than or equal to N_α. Finally we show that we can extend the N _α to ⊃_α in the second theorem consistently with ZFC and while having N_i« ⊃_i for 0 < i ≤ α. Similar results hold for arbitrary ordinals α with |α| = κ and L_κ+.ω.