### The amalgamation spectrum

by Baldwin and Kolesnikov and Shelah. [BKSh:927]

J Symbolic Logic, 2009

For every natural number k^*, there is a class
boldmath {K}_* defined by a sentence in
L_{omega_1, omega} that has no models of cardinality
> beth_{k^*+1}, but mbox {boldmath {K}}_* has the d
isjoint amalgamation property on models of cardinality
<= aleph_{{k^*}-3} and has models of cardinality
aleph_{{k^*}-1} . More
strongly, For every countable ordinal alpha^*, there
is a class mbox {boldmath {K}}_* defined by a sentence in
L_{omega_1, omega} that has no models of cardinality
> beth_{alpha}, but mbox {boldmath {K}}_* has the
disjoint amalgamation property on models of cardinality
<= aleph_{alpha} .
Similar results hold for arbitrary kappa and
L_{kappa^+, omega} .

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