Hamiltonian is the proper adjective for Hamilton.
The proper adjective for southwest is "southwestern."
The proper adjective for Haiti is Haitian.
The proper adjective for Portuguese is "Portuguese."
Vietnamese is the proper adjective for Vietnam.
The proper adjective for George is Georgian.
The proper adjective for Siam is Siamese.
Freudian is the proper adjective for Freud.
The proper adjective for Georgia is Georgian.
The proper adjective for Mayan is "Mayan."
No, old is not a proper adjective. A proper adjective is an adjective derived from a proper noun, such as Swiss cheese, Bolivian pottery, Chinese silk, etc.
The proper adjective for Antigua is Antiganantigan