indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
A class of structures in model theory has the amalgamation property if for any three structures $A,B,C$ and embeddings $f_B: A\hookrightarrow B$, $f_C: A\hookrightarrow C$, there exist embeddings $g_B:B\hookrightarrow D$ and $g_C: C\hookrightarrow D$ such that $g_B\circ f_B = g_C\circ f_C$.
One of the simplest cases is when the free amalgam of structures $B\oplus_A C$ exists.
The amalgamation method for generating strongly minimal theories is introduced in
62(2):147–166, 1993. Stability in model theory, III (Trento, 1991).
Recent references include
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