Finite-tuple homogeneity
= Finite-tuple homogeneity
A <first-order structure> has <finite-tuple homogeneity> when any two finite <tuples> of the same parameter-free <complete type> are related by an automorphism. For countably infinite <first-order structures> this agrees with extension of finite <partial elementary maps> to automorphisms. <Countable> <atomic models> have this property by a <back-and-forth method> using isolating <logical formulas>. Uncountable <atomic models> can fail it.