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 Kk defined by a sentence in Lω₁,ω that has no models of cardinality greater than ℶk+1, but Kk has the disjoint amalgamation property on models of cardinality less than or equal to ℵk-3 and has models of cardinality ℵ{k}-1. More strongly, we can have disjoint amalgamation up to ℵα for α < ω₁, but have a bound on size of models. Theorem B For every countable ordinal α, there is a class Kα defined by a sentence in Lω₁,ω that has no models of cardinality greater than ℶω₁, but K does have the disjoint amalgamation property on models of cardinality less than or equal to ℵα. Finally we show that we can extend the ℵα to ℶα in the second theorem consistently with ZFC and while having ℵi≪ ℶi for 0< i≤ α. Similar results hold for arbitrary ordinals α with |α|=κ and Lκ⁺,ω. THEOREM A For every natural number k, there is a class K(k) defined by a sentence in L(omega vertical bar omega) that has no models of cardinality greater than superset of(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-1) More strongly. we can have disjoint amalgamation up to N(a) for alpha < omega(1), but have a bound oil size of models. THEOREM B For every countable ordinal alpha. there is it class K(alpha) defined by a sentence in L(omega 1 vertical bar omega) that has no models of cardinality greater than superset of(omega 1), but K does have the disjoint amalgamation property on models of cardinality less than or equal to N(alpha). Finally we show that we can extend the N(alpha) to superset of(alpha) in the second theorem consistently with ZFC and while having Ni << superset of(i) for 0 < i <= alpha. similar results hold for arbitrary ordinals alpha with vertical bar alpha vertical bar = kappa and L(kappa)+,(omega)