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It is well known that the classic {\L}o\'s-Tarski preservation theorem failsin the finite: there are first-order definable classes of finite structuresclosed under extensions which are not definable (in the finite) in theexistential fragment of first-order logic. We strengthen this by constructingfor every $n$, first-order definable classes of finite structures closed underextensions which are not definable with $n$ quantifier alternations. Theclasses we construct are definable in the extension of Datalog with negationand indeed in the existential fragment of transitive-closure logic. Thisanswers negatively an open question posed by Rosen and Weinstein.

Extension Preservation in the Finite and Prefix Classes of First Order Logic