The Ordered conjecture of Kolaitis and Vardi asks whether fixed-point logic differs from first-order logic on every infinite class of finite ordered structures. In this paper, we develop the tool of bounded variable element types, and illustrate its application to this and the original conjectures of McColm, which arose from the study of inductive definability and infinitary logic on proficient classes of finite structures (those admitting an unbounded induction). In particular, for a class of

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... inite structures, we introduce a compactness notion which yields a new proof of a ramified version of McColm's second conjecture. Furthermore, we show a connection between a model-theoretic preservation property and the Ordered Conjecture, allowing us to prove it for classes of strings (colored orderings). We also elaborate on complexity-theoretic implications of this line of research. Comments Abstract. The Ordered conjecture of Kolaitis and Vardi asks whether xed-point logic di ers from rst-order logic on every in nite class of nite ordered structures. In this paper, we d e v elop the tool of bounded variable element t ypes, and illustrate its application to this and the original conjectures of McColm, which arose from the study of inductive de nability and in nitary logic on pro cient classes of nite structures (those admitting an unbounded induction). In particular, for a class of nite structures, we i n troduce a compactness notion which yields a new proof of a rami ed version of McColm's second conjecture. Furthermore, we show a connection between a model-theoretic preservation property and the Ordered Conjecture, allowing us to prove it for classes of strings (colored orderings). We also elaborate on complexity-theoretic implications of this line of research.