Fraïssé's theorem

In mathematics, Fraïssé's theorem, named after Roland Fraïssé, states that a class K of finite relational structures is the age of a countable homogeneous relational structure if and only if it satisfies the following four conditions:

• K is closed under isomorphism;
• K is closed under taking induced substructures;
• K has only countably many members up to isomorphism;
• K has the amalgamation property.

If these conditions hold, then the countable homogeneous structure whose age is K is unique up to isomorphism.^[1]

Fraïssé proved the theorem in the 1950s.

References

1. Meenaxi Bhattacharjee, Dugald Macpherson, Rögnvaldur G. Möller, Peter M. Neumann. Notes on Infinite Permutation Groups. Lecture Notes in Mathematics 1689. 1998: Springer. pp. 155–158. ISBN 978-3-540-64965-6.