the amalgamation property

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Definitions:

  • The age of a structure $M$ is the class of finitely generated substructures of $M$.
  • A class of structures K has the amalgamation property (AP) if

Whenever $A,B,C$ belong to $K$ and $e:A \to B$, $f:A \to C$ are embeddings, then there is a $D$ belonging to $K$ and embeddings $g:B \to D$, $h:C \to D$ such that $g \circ e = h \circ f$.

Give a direct argument showing that the age of the complete (undirected) graph $K_n$ satisfies the amalgamation property.