Automatic structures are—possibly infinite—structures which are
finitely presentable by means of finite automata on strings or trees.
Largely motivated by the fact that their first-order theories are uniformly
decidable, automatic structures gained a lot of attention in the "logic in
computer science" community during the last fifteen years. This thesis
studies the model-theoretic complexity of automatic linear orders in terms
of two complexity measures: the finite-condensation rank and the Ramsey
degree. The former is an ordinal which indicates how far a linear order is
away from being dense. The corresponding main results establish optimal
upper bounds on this rank with respect to several notions of automaticity.
The Ramsey degree measures the model-theoretic complexity of well-orders by
means of the partition relations studied in combinatorial set theory. This
concept is investigated in a purely set-theoretic setting as well as in the
context of automatic structures.