Some new computable structures of high rank

Abstract

We give several new examples of computable structures of high Scott rank. For earlier known computable structures of Scott rank ω 1 C K \omega _1^{CK} , the computable infinitary theory is ℵ 0 \aleph _0 -categorical. Millar and Sacks asked whether this was always the case. We answer this question by constructing an example whose computable infinitary theory has non-isomorphic countable models. The standard known computable structures of Scott rank ω 1 C K + 1 \omega _1^{CK}+1 have infinite indiscernible sequences. We give two constructions with no indiscernible ordered triple.