Failure of the Los-Tarski preservation theorem for the fluted fragment

Abstract

The classical Los-Tarski theorem characterises first-order sentences preserved under extensions as the existentially definable ones. In [6], Purdy claimed that the analogous preservation theorem holds for the fluted fragment. We refute this claim by constructing, over an equality-free vocabulary with only one binary relation symbol, a fluted sentence of quantifier rank three which is preserved under extensions but is not equivalent, even over finite structures, to any existential fluted sentence.